BUS REAL-TIME ARRIVAL PREDICTION USING STATISTICAL PATTERN RECOGNITION TECHNIQUE
By
Nam Hoai Vu, M.Sc., (2000) Hanoi University of Civil Engineering, Vietnam
A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
Department of Civil and Environmental Engineering Carleton University Ottawa, Ontario, Canada
© December 2006 Nam Hoai Vu
The Doctor of Philosophy in Civil Engineering is a joint program with the University of Ottawa, administrated by the Ottawa-Carleton Institute for Civil Engineering
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Library and Bibliotheque et Archives Canada Archives Canada
Published Heritage Direction du Branch Patrimoine de I'edition
395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada
Your file Votre reference ISBN: 978-0-494-23303-0 Our file Notre reference ISBN: 978-0-494-23303-0
NOTICE: AVIS: The author has granted a non L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non sur support microforme, papier, electronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats.
The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in et des droits moraux qui protege cette these. this thesis. Neither the thesis Ni la these ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent etre imprimes ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission.
In compliance with the Canadian Conformement a la loi canadienne Privacy Act some supporting sur la protection de la vie privee, forms may have been removed quelques formulaires secondaires from this thesis. ont ete enleves de cette these.
While these forms may be included Bien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. i * i Canada Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract
Given the realization that real-time bus arrival information is viewed positively by
passengers of public transit, many bus transit agencies of various sizes are developing
Real-time Bus Arrival Information System (RETBAIS) following the implementation of
Automatic Vehicle Location (AVL) and Automatic Passenger Counter (APC) systems.
This research focuses on one important element of the RETBAIS, the real-time prediction
model. Data required for the research were retrieved from the APC and AVL systems of
the City of Ottawa/OC Transpo. The developed model has two main modules: Running
Time Prediction Module (RTM) and Dwell Time Prediction Module (DTM).
The RTM is based on the statistical pattern recognition methodology. Given a
pattern defining bus running time being predicted, the trained RTM automatically
searches through the historical patterns which are the most similar to the new pattern and
based on that, the prediction of a bus running time is made. The RTM was tested with
different data sets of various bus running time situations. It was found that it worked
well as indicated by the average relative prediction error of as low as 5% for the
Transitway route and about 7% for the mixed-traffic bus route. Moreover, this module
performed in a consistent manner even when unusual bus operational scenarios were
used.
The DTM has four sub-modules. The first two sub-modules are also based on a
recognition technique for predicting separately the number of passengers boarding and
alighting. The third sub-module is used to examine the relationship between actual dwell
times and various explanatory variables. The last one is based on the fact that passengers
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. choose the most convenient door for boarding and alighting. Having tested with datasets,
the DTM proved that it can predict passenger activities with satisfactory accuracy without
any specific prior assumptions on the complicated relationship between dwell time and
the influencing factors.
When the constituent modules are integrated, the whole model can predict bus
arrival times at every downstream stop. The prediction accuracy increased with new data
availability. The average relative prediction error varied from 3 to 8%.
In order to provide bus dispatchers with tools for managing bus fleet, two
methods to detect bus on-time performance and bus bunching were developed. By using
these tools, a bus dispatcher can easily know ahead of time if the bus is on-time, late,
early, or bunching is likely to occur.
By offering fast, accurate and reliable predictions, it is contended that the
developed real-time prediction model will enhance the bus arrival information system
and therefore will be a contribution to public transportation operation.
ii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments
I would like to express my deepest gratitude to Prof. Ata. M. Khan for
encouragement, patience, support and invaluable scientific guidance in supervising this
thesis.
I am grateful to the staffs of the OC Transpo (Ottawa, Canada) for several
discussions, comments and data provisions on the various aspects of this thesis. Special
thanks to Mr. Joe Koffman, Mr. Brian Barclay and Ms. Sylvie Paquette. I would like to
say thank you to Mr. Kean Lew and Mrs. Stephen Hotard (PTV America Inc.) for helping
me to use the VISSIM software.
I am greatly thankful to Prof. Yasser Hassan, Prof. William Johnson, and Prof.
Steven Prus for valuable suggestions on this thesis research.
Financial support by the Vietnamese Government is gratefully acknowledged.
I want to thank my friends; Mr. Phung Viet Anh who was always willing to help
me during difficulties; Mr. Jarbar Siddique for interesting conversations in the common
favorite area of bus transit; Ms. Sandra Majkik who shared data with me.
I am so indebted to my wife Mrs. Huyen Vu and to my son Hieu Vu for
continuous encouragements, patience, sacrifice and their love. I love you both.
Four years of living and studying in this country tattooed in my mind about a
beautiful country with clement people. Thank you Canada!
Nam Hoai Vu
m
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS
ABSTRACT...... i ACKNOWLEDGMENTS...... iii TABLE OF CONTENTS...... iv LIST OF TABLES...... xi LIST OF FIGURES...... xv ABBREVIATIONS...... xviii
CHAPTER 1: INTRODUCTION
1.1 Overview ...... 1 1.2 Background ...... 4 1.3 Problem Statement ...... 6 1.4 Goals and Objectives ...... 7 1.5 Study Methodology ...... 8 1.6 Thesis Document and Organization ...... 11
CHAPTER 2: LITERATURE REVIEW
2.1 Introduction ...... 14 2.2 Real-Time Bus Arrival Information System: Current State of Development... 14
2.2.1 AVL System and APC System ...... 15 2.2.1.1 Automatic Vehicle Location System ...... 15 2.2.1.2 Automatic Passenger Counting System ...... 19 2.2.1.3 Uses of Retrieved AVL-APC data in RETBAIS ...... 21 2.2.2 Bus Running Time Prediction Algorithms ...... 25 2.2.2.1 Blacksburg (Virginia) Prediction Algorithms...... 25 22.2.2 Portland (Oregon) and King County Metro, Seattle (Washington) Prediction Algorithm ...... 26
iv
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2.2.3 Texas Transportation Institute Algorithms ...... 28 2.2.2.4 New Jersey University Algorithm ...... 28 2.2.2.5 University of Toronto Algorithm ...... 29 2.2.2.6 Artificial Neural Networks (ANN)-Based Algorithms ...... 30 2.2.2.7 Classical Statistical Regression Models ...... 31 2.2.2.8 Other Prediction Models ...... 33 2.2.3 Bus Dwell Time Prediction Models ...... 35 2.2.4 Media and Communication System ...... 38 2.2.4.1 Information Dissemination Media ...... 39 2.2.4.2 Communication System ...... 40
2.3 Summary...... 40
CHAPTER 3: MECHANISM OF THE PROPOSED MODEL AND AVL -APC DATA COLLECTION
3.1 Introduction ...... 42 3.2 Component of Bus Trip Time and Influencing Factors ...... 42 3.2.1 Actual Moving Time ...... 44 3.2.2 Dwell Time...... 45 3.2.3 Traffic Signal Delay ...... 46 3.2.4 General Delay ...... 46 3.2.5 Recovery Time ...... 48 3.3 Proposed Structures and Components ...... 48 3.3.1 General Discussion ...... 48 3.3.2 Assumptions ...... 50 3.3.3 Structure and the Building Modules of the Proposed Model ...... 51 3.3.4 Mechanism of the Proposed Model ...... 53 3.4 APC and AVL Data Collection ...... 59 3.4.1 Bus Route Selection ...... 60 3.4.2 Data Sets...... 62
v
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4.3 APC and AVL Data Description ...... 64 3.4.4 Data Characteristics ...... 66 3.4.4.1 Arrival Time...... 66 3.4.4.2 Dwell Time and Passenger Activities ...... 76 3.5 Summary...... 82
CHAPTER 4: BUS RUNNING TIME PREDICTION MODULE
4.1 Introduction ...... 83 4.2 Problem Statement ...... 84 4.3 Nonparametric Regression Methods in Statistical Pattern Recognition ...... 85 4.3.1 LOWESS Estimation Method ...... 87 4.3.1.1 General Functions ...... 87 4.3.1.2 Bandwidth Selection ...... 89 4.3.1.3 Weighting Kernel Function Selection ...... 93 4.3.14 Degree of Polynomial Regression ...... 93 4.4 Modelling Bus Running Time Prediction by using LOWESS Method and APC -AVL Data...... 94 4.4.1 APC and AVL data ...... 94 4.4.2 Parameter Selection ...... 94 4.4.3 Pattern Selection ...... 95 4.4.4 Pattern Recognition ...... 96 4.4.4.1 Euclidean Distance Calculation ...... 96 4.4.4.2 Optimal Bandwidth by Leave-one-out Method ...... 96 4.4.4.3 Recognition of the Neighbours ...... 100 4.4.5 Prediction ...... 100 4.4.6 Update Prediction ...... 102 4.5 Summary...... 103
vi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5: BUS DWELL TIME PREDICTION MODULE
5.1 Introduction ...... 104 5.2 Dwell Time Prediction Module ...... 105 5.2.1 Discussions on Previous Works ...... 105 5.2.2 Dwell Time Prediction Module (DTM) ...... 108 5.2.3 Real-Time Boarding Passenger Prediction Sub-module ...... 110 5.2.3.1 Parameter Selection ...... 110 5.2.3.2 Pattern Selection ...... 110 5.2.3.3 Pattern Recognition ...... 114 5.2.3.4 Prediction ...... 114 5.2.3.5 Update Prediction ...... 115 5.2.4 Real-time Alighting Passenger Prediction Sub-module ...... 115 5.2.4.1 Parameter Selections ...... 115 5.2.4.2 Pattern Selection ...... 116 5.2.4.3 Pattern Recognition ...... 116 5.2.4.4 Prediction ...... 117 5.2.4.5 Update Prediction ...... 117 5.2.5 Regression Sub-Module ...... 118 5.2.5.1 Variables Selection and Preparation ...... 118 5.2.5.2 Regression Functions ...... 120 5.2.6 Busiest Door Prediction Sub-Module ...... 127 5.2.6.1 Rigid-Body Bus ...... 128 5.2.6.2 Articulated Bus...... 136 5.2.7 Method A vs. Method B and the Selection ...... 142 5.2.7.1 Method A ...... 143 5.2.1.2 Method B ...... 143 5.2.13 Rigid-body Bus ...... 144 5.2.7.4 Articulated Bus...... 144 5.2.7.5 Accuracy Performance of the Two Methods ...... 146 5.3 Summary...... 147
vii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6: MODEL PERFORMANCE
6.1 Introduction ...... 149 6.2 Computer Program Development ...... 150 6.3 Simulation of Bus Operation Scenarios ...... 150 6.3.1 The VISSIM Simulator ...... 151 6.3.2 Transit Route Coding in VISSIM ...... 153 6.3.3 Calibration and Validation ...... 155 6.3.4 Bus Operation Scenarios, Micro-simulation Runs, and the VISSIM Outputs ...... 157 6.4 Model Testing and Comparison ...... 162 6.4.1 Evaluation Criteria for Prediction Performance ...... 162 6.4.2 Reference Predictors ...... 163 6.4.2.1 The Naive Model ...... 164 6.4.2.2 The Kalman Filter- Based Model ...... 164 6.4.3 Testing the Developed Model and the Reference Predictors ...... 166 6.4.3.1 Data Issues...... 166 6.4.3.2 Running Time Prediction Performance ...... 167 6.4.3.3 Boarding Passenger Prediction Performance ...... 176 6.4.3.4 Alighting Passenger Prediction Performance ...... 178 6.4.3.5 Prediction Performance with Actual Data ...... 180 6.4.3.6 Tukey Test for Performance Comparison ...... 184 6.4.3.7 Bus Arrival Time Prediction Performance ...... 186 6.5...... Summary...... 188
CHAPTER 7: REAL-TIME PREDICTION INTERVAL, ON-LINE SCHEDULE ADHERENCE EVALUATION AND BUS BUNCHING DETECTION
7.1 Introduction ...... 190 7.2 Real-time Prediction Interval ...... 190 7.3 On-line Schedule Adherence Evaluation ...... 194
viii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.3.1 Methodology ...... 194 7.3.2 A Numerical Example...... 196 7.3.3 Discussion ...... 201 7.4 Real-time Bus Bunching Detection Method ...... 203 7.4.1 Methodology ...... 203 7.4.2 A Numerical Example ...... 206 7.5 Further Possible Applications of the Developed Model ...... 207 7.6 Real-time Bus Arrival Broadcasting ...... 208 7.7 Level of Spatial and Temporal Detail of Data for the Developed Model ...... 209 7.8 Summary ...... 210
CHAPTER 8: CONCLUSIONS AND RECOMMENDATIONS
8.1 Conclusions...... 211 8.2 Recommendations and Future Research ...... 216
REFERENCES...... 219 APPENDICIES...... 234 Appendix A ...... 235 Appendix B1 ...... 237 Appendix B2 ...... 243 Appendix B3 ...... 246 Appendix C ...... 249 Appendix D1 ...... 258 Appendix D2 ...... 267 Appendix D3 ...... 281 Appendix D4 ...... 285 Appendix D5 ...... 317 Appendix D6 ...... 330 Appendix D7 ...... 341 Appendix D8 ...... 382
ix
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix D9 384
x
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES
Table 2.1 Transit Agencies with Real-time Bus Arrival Information System ...... 23
Table 3.1 Factors that influence Bus Running Time and Delay ...... 43
Table 3.2 Distance between stops BLAIR and LEBRETON. Bus Route 95 (Orleans-Nepean South) ...... 61
Table 3.3 Distance between stops GREENBORO and RIDEAU. Bus Route 1 (South Keys - Ottawa Rockliffe) ...... 62
Table 3.4 Data Sets for Bus Route 95 and Bus Route 1...... 63
Table 3.5 Data Collectable by APC and AVL Systems ...... 65
Table 3.6 Means and Standard Deviations of Bus on-time Performance in the Winter Bus Route 1 (Greensboro- Rideau) ...... 67
Table 3.7 Means and Standard Deviations of Bus on-time Performance in the Summer Bus Route l(Greensboro-Rideau) ...... 68
Table 3.8 Means and Standard Deviations of Bus on-time Performance in the Winter. Bus Route 95(Blair- Lebreton) ...... 68
Table 3.9 Means and Standard Deviations of Bus on-time Performance in the Summer . Bus Route 95(Blair-Lebreton) ...... 69
Table 3.10 Two-sample t statistics for Winter and Summer Time. Route 95- Morning period ( 8 a.m. to 10 a.m.) ...... 70
Table 3.11 Two-sample t statistics for Winter and Summer Time. Route 95- Noon period (12 a.m. to 2 p.m.) ...... 70
Table 3.12 Two-sample t statistics for Winter and Summer Time Route 95- Afternoon period (4 p.m. to 6 p.m.) ...... 71
Table 3.13 Two-sample t statistics for Winter and Summer Time Route 1- Morning period ( 8 a.m. to 10 a.m.) ...... 72
Table 3.14 Two-sample t statistics for Winter and Summer Time Route 1- Noon period (12 a.m. to 2 p.m.) ...... 72
xi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.15 Two-sample t statistics for winter and summer time Route 1- Afternoon period (4 p.m. to 6 p.m.) ...... 72
Table 3.16 Two-sample t statistics for Different Time Periods of Day. Route 95- the winter time (the WINTER data set) ...... 73
Table 3.17 Two-sample t statistics for Different Time Periods of Day. Route 95- the summer time (the SUMMER data set) ...... 74
Table 3.18 Two-sample t statistics for Different Time Periods of Day. Route 1- the winter time (the WINTER data set) ...... 74
Table 3.19 Two-sample t statistics for Different Time Periods of Day. Route 1- the summer time (the SUMMER data set) ...... 75
Table 3.20 Descriptive Statistics of Passenger Activities. Route 1- time period (8 a.m. to 10 a.m.) ...... 79
Table 3.21 Descriptive Statistics of Passenger Activities. Route 1- time period (12 a.m. to 2 p.m.) ...... 80
Table 3.22 Descriptive Statistics of Passenger Activities Route 1- time period (4 p.m. to 6 p.m.) ...... 80
Table 3.23 Descriptive Statistics of Passenger Activities Route 95- time period (8 a.m. to 10 a.m.) ...... 80
Table 3.24 Descriptive Statistics of Passenger Activities. Route 95- time period (12 a.m. to 2 a.m.) ...... 81
Table 3.25 Descriptive Statistics of Passenger Activities. Route 95- time period (4 p.m. to 6 p.m.) ...... 81
Table 4.1 Common Kernel Functions for Univariate Data ...... 93
Table 5.1 Variable Selections for RESM ...... 119
Table 5.2 Coefficients of the Best Regression ...... 123
Table 5.3 Model Summary for the Best Regression ...... 124
Table 5.4 Proposed Non-linear Regression Models ...... 126
Table 5.5 Validated Dataset for Logistic Regressions ...... 132
Table 5.6 Best Binary Logistic Regression Equations ...... 133
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 5.7 Model Summary of Type B-L.l ...... 133
Table 5.8 Prediction Performance of Type BL-1 ...... 134
Table 5.9 Coefficients of Type BL-1 ...... 135
Table 5.10 The Best Multinomial Logistic Regression Equations ...... 139
Table 5.11 Likelihood Ratios of Type B-ML.3 ...... 140
Table 5.12 Parameters of Type B-ML.3 ...... 141
Table 5.13 Measures of Accuracy ...... 143
Table 5.14 Average Service Time Selection(s/p) ...... 144
Table 5.15 Example Output of Dwell time Prediction of Method A and Method B Applied for Rigid-body Bus ...... 145
Table 5.16 Example Output of Dwell time Prediction of Method A and Method B Applied for Articulated Bus ...... 146
Table 5.17 Mean Absolute Prediction Error of Method A vs. Method B ...... 147
Table 6.1 Differences between VISSIM( version 4.1) and CORSIM (TSIS 5.1) for public transit simulation ...... 152
Table 6.2 Kinetic Characteristics of the OC bus ...... 153
Table 6.3 VISSIM Calibrated Parameters ...... 157
Table 6.4 Dimensionality and Required Sample Size ...... 159
Table 6.5 Scenario Design of Simulation ...... 160
Table 6.6 Accuracy Criteria for Model Testing ...... 163
Table 6.7 Cross-validation MAPE, MRE, and RMSE ...... 169
Table 6.8 Route 1- Running Time Prediction with Overall Situations: MAPE, MRE, and RM SE ...... 171
Table 6.9 Studentized Range of Test Sets ...... 185
Table 6.10 Result of Tukey’s Procedure at Significance Level =0.05 ...... 185
xiii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 6.11 Holmwood Stop: Bus Arrival Time Prediction Performance of the 3 predictors ...... 186
Table 6.12 Gladstone Stop: Bus Arrival Time Prediction Performance of the 3 predictors ...... 187
Table 6.13 Rideau Stop: Bus Arrival time prediction performance of the 3 predictors ...... 187
Table 6.14 Numerical Examples of Real-time Bus Arrival Prediction Using Developed Models for 3 full trips (seconds) ...... 188
Table 7.1 Real-time Prediction Interval for Bus Running Time of Route 1 (Time in second) ...... 194
Table 7.2 Goodness-of-fit test Summary (a =0.05) ...... 198
Table 7.3 Parameters of the Two Consecutive Buses ...... 206
Table 7.4 Level of Spatial and Temporal Details for Data Capture ...... 209
xiv
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES
Figure 1.1 Research Framework ...... 9
Figure 2.1 Signpost-based Automatic Vehicle Location ...... 18
Figure 2.2 GPS-base Automatic Vehicle Location ...... 18
Figure 2.3 The APC System Mounted on a Bus ...... 19
Figure 2.4 AVL and APC Systems on a Bus ...... 21
Figure 3.1 Bus Trip Time Components ...... 43
Figure 3.2 A Simplified Sketch of Bus Operation ...... 51
Figure 3.3 Process of Real-time Prediction ...... 54
Figure 3.4A Bus Route 95 (Blair- Lebereton) ...... 61
Figure 3.4B Bus Route 1 (South Keys-Rideau) ...... 62
Figure 3.5 Structures of APC Data at pointcheck ...... 64
Figure 3.6 Structure of APC Data on Passenger Activity ...... 66
Figure 3.7 Dwell time Histogram. Route 1- Time period (8 a.m. tolO a.m.) ...... 76
Figure 3.8 Dwell time Histogram. Route 1- Time period (12 a.m. to 2 p.m.) ...... 77
Figure 3.9 Dwell time Histogram. Route 1- Time period (4 p.m. to 6 p.m.) ...... 77
Figure 3.10 Dwell time Histogram. Route 1- Time period (4 p. m. to 6 p.m.) ...... 77
Figure 3.11 Dwell time Histogram. Route 95 - Time period (8 a.m. to 10 a.m.) ...... 78
Figure 3.12 Dwell time Histogram. Route 95 - Time period (12 a.m. to 2 p.m.) ...... 78
Figure 4.1 Bandwidth Selection and NP Regression ...... 90
Figure 4.2 Constant Bandwidth Problem ...... 90
Figure 4.3 N-Nearest Type of Bandwidth ...... 91
Figure 4.4 Automatic Process of Searching Optimal Number of Neighbours 97
xv
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5.1 Dwell time Prediction Module. Method A ...... 109
Figure 5.2 Dwell time Prediction Module. Method B ...... 109
Figure 5.3 Boxplot of Dwell time. Bus route 1 (All seasons and all time dataset) ...... 121
Figure 5.4 Boxplot of validated dwell time. Bus route 95 (All seasons and all time dataset) ...... 121
Figure 6.1 Structure and Constituent Functions of the Computer Program ...... 150
Figure 6.2 Bus Routes 95 and 1 coded in VISSIM 4.10 ...... 154
Figure 6.3 Vehicle Record Configuration Tool in the VISSIM ...... 155
Figure 6.4 Route 95- Running Time Prediction: MAPE and MRE ...... 169
Figure 6.5 Route 95- Running Time Prediction: RMSE ...... 170
Figure 6.6 Route 1- Running Time Prediction with Overall Situation: MAPE and MRE ...... 171
Figure 6.7 Route 1- Running Time Prediction with Overall Situation: RMSE...... 171
Figure 6.8 Routel-The Differences of MAPE Between Overall and Slowdown zone Situations...... 173
Figure 6.9 Route 1 -The difference of RME Between Overall and Slowdown zone Situations...... 173
Figure 6.10 Routel-The Difference of RMSE Between Overall and Slowdown zone Situations...... 173
Figure 6.11 Nearest Neighbour and Cross-Validation Least-square Error (CR) for Bus Running Time Prediction. Case 43-2200, Route 1 ...... 175
Figure 6.12 Nearest Neighbour and Cross-Validation Least-square Error (CR) for Bus Running Time Prediction. Case 54-2200, Route 95 ...... 175
Figure 6.13 Route 95- Boarding Passenger Prediction: MAPE and MRE ...... 177
Figure 6.14 Route 95- Boarding Passenger Prediction: RMSE ...... 177
Figure 6.15 Nearest Neighbour and Cross-Validation Least-square Error (CR)
xvi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for boarding passenger prediction. Case 24-1200, Route 95 ...... 178
Figure 6.16 Route 95- Alighting Passenger Prediction: MAPE and MRE ...... 179
Figure 6.17 Route 95- Alighting Passenger Prediction: RMSE ...... 179
Figure 6.18 Nearest Neighbour and Cross-Validation Least-square Error (CR) for Alighting passenger Prediction. Case 20-1200, Route 95 ...... 180
Figure 6.19 Route 95-Running Time Prediction with Actual Data: MAPE and MRE ...... 181
Figure 6.20 Route 95-Running Time Prediction with Actual data: RMSE ...... 181
Figure 6.21 Route 1- Running Time Prediction with Actual Data: MAPE and MRE ...... 181
Figure 6.22 Route 1-Running Time Prediction with Actual Data: RMSE ...... 182
Figure 6.23 Route 95- Boarding Passenger Prediction with Actual Data MAPE and MRE ...... 182
Figure 6.24 Route 95- Boarding Passenger Prediction with Actual Data: RMSE.... 183
Figure 6.25 Route 95- Alighting Passenger Prediction with Actual Data MAPE and MRE ...... 183
Figure 6.26 Route 95- Alighting Passenger Prediction with Actual Data: RMSE.... 183
Figure 7.1 Fitted Density for OAP at Gladstone Stop-Routel ...... 200
Figure 7.2 On-time Limits and Predictions Interval ...... 201
Figure 7.3 Tendency of Bus Bunching P> 0 ...... 204
Figure 7.4 Tendency of Bus Bunching P = 0 ...... 204
xvii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABBREVIATIONS
ANN Artificial Neural Networks
APC Automatic Passenger Counter
APTS Advanced Public Transportation System
AVL Automatic Vehicle Location
ATIS Advanced Traveler Information System
BDSM Busiest- door Prediction Sub-Module
CBD Central Business District
CORSIM Corridor Simulation
CV Cross- Validation
CR Cross-validation Least-square Error
DGPS Differential Global Positioning System
DMS Dynamic Message Sign
DR Dead Reckoning
DTM Dwell Time Prediction Module
FTA Federal Transit Administration
GPS Global Positioning System
GIS Geographic Information System
OAP Off-line Adherence Performance
OC Transpo Ottawa-Carleton Regional Transit Commission
OLS Ordinary Least Square
PBSM Punctuality and Bunching Prediction Sub-Module
PDA Personal Digital Assistant
xviii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ITS Intelligent Transportation System
LORAN-C C configuration o f Long Range Navigation
LOWESS Locally Weighted Scatter Plot Smoothing
MAPE Mean Absolute Prediction Error
NP Non-Parametric Regression
RASM Real-time Alighting Passenger Prediction Sub-Module
RAP Real-time Adherence Performance
RBSM Real-time Boarding Passenger Prediction Sub-Module
RMSE Root Mean Square Error
RETBAIS Real-Time Bus Arrival Information System
RESM Regression Sub- Module
RTM Running time Prediction Module
SPS Standard Positioning System
SO Signpost and Odometer
SP Recognition Statistical Pattern Recognition
Tri-Met Tri-County Metropolitan Transportation District o f Oregon
VISSIM Verkerhr In Stadten-SIMulation
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1
INTRODUCTION
1.1 Overview
Many cities around the world are struggling with transportation problems.
Managing congestion, reducing air and noise pollution, minimizing fuel consumption,
maintaining adequate supply to serve an increasing demand for travel, and preserving
community quality of life, are always the challenges for the cities to face. New or
widened highways are no longer the preferred solutions of today due to fiscal constraints,
conservation of land and petroleum fuels, and environmental quality objectives.
To solve the transportation problems, the development of a convenient and
modem public transportation network in the city is usually considered as a first priority
solution by the city authorities. Enhancing customer satisfaction by improving service
quality will be a key to the success in retaining existing riders and diverting users of
private automobile to the public transportation system. In America, if all people who are
riding transit to work drove instead, their cars would be as long as 23,000 miles line of
traffic, long enough to circle the earth (FTA 1996 Report).
Innovations in communication networks, navigation systems, Global Positioning
System (GPS), Geographic Information System (GIS), sensor technologies, the Internet,
and computer software, are contributing to the enhancement of the public transportation
1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2
system. At present, the Intelligent Transportation Systems (ITS) in various countries are
the vivid examples of the successes in applying new technologies in the transportation
field.
The advanced public transportation system (APTS) is one of the main
components in the ITS architecture in many countries. The APTS implemented by GPS,
GIS, Automatic Vehicle Location System (AVL), and Automatic Passenger Counter
System (APC), enables transit agencies to capture the real-time information including
vehicle location, speed, operational status, and passenger occupancy. Real-time
information is a valuable and reliable input for transit passenger information systems to
assist transit providers in planning, operation and management.
Although various transit agencies have implemented new technologies to collect
real-time information about their en-route bus fleets, they have been facing difficulties in
disseminating reliable bus arrival information to the customer. Traffic conditions reflect
stochastic processes. Therefore, they influence a bus adherence to schedule in a random
way. It is a challenge for transit agencies to know the arrival time ahead of time. Only the
real-time predictions based on real-time information are capable of attracting customers
to the public transit and the use of the predicted arrival information can potentially
influence the daily activities of travellers.
The APC systems have been deployed in some transit agencies in North America
(the United States and Canada) since the 1980s. The time when the first AVL system was
launched was even earlier. In 1969, Chicago (USA), implemented an AVL system and
Toronto, (Canada) installed one in 1972. The names of AVL and APC systems explain
their functions. The AVL system provides dynamically the location, speed, and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3
operational status of a remote bus while an APC system is able to transmit the
information about passenger boarding and alighting at stops, running time between stops,
dwell times, and several service characteristics to the control center. The AVL and APC
systems provide much data which can potentially serve as inputs for the development of
bus arrival prediction models.
When the first AVL system was initiated in Chicago, a 10-year vision of using
AVL systems to improve the schedule performance, was described optimistically by
Klopfenstein, (1986, p.3). Since that time, indeed, many transit agencies still do not
operate AVL and APC systems. As of 2000, only 88 out of 548 transit agencies in the
United States were operating AVL system and 145 others were planning AVL systems
(Schweiger, 2003). A small fraction of the 88 agencies are providing real-time arrival
information to the customer via Advanced Traveller Information System (ATIS) accessed
by passengers at home, work places, and bus stations via the World Wide Web (WWW)
Internet system or wireless devices such as internet capable cell phones, Palm Pilots, and
Personal Digital Assistants (PDAs). The reason behind the delay is not only that an AVL
system is still a significant investment but also the agencies are waiting for the most
preferable real-time information system that is capable of providing bus real-time arrival
information, workable with the existing technologies and infrastructure, and suitable for
their fiscal conditions.
The number of transit agencies using APC systems was even fewer. A survey
conducted by Daniel (1998) showed that out of the 27 US transit agencies in the survey,
only five agencies use the APC system. Almost all of these agencies used pencil and
manual check for different purposes. The Ottawa-Carleton Regional Transit Commission
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4
(OC Transpo), an 827-bus system serving the capital city of Ottawa (Canada), is a
pioneer and long-time user of the APC system. Over the years, data collected from the
APC system and the signpost-based AVL system are serving the OC Transpo in terms of
increasing the quality of service, monitoring the average loads and passenger activities,
and maintaining schedule adherence. However, like many other transit agencies, the OC
Transpo has not yet deployed a real-time bus arrival information system or RETBAIS for
short.
Clearly, there is a need to develop models that can utilize the huge and valuable
AVL-APC database for predicting bus arrival times in order to improve the quality of the
arrival information disseminated to passengers.
1.2 Background
Transit agencies provide bus information to passengers in different ways. These
vary from traditional methods such as printed schedules to modem methods such as
arrival times on an electronic sign, also known as Dynamic Message Sign (DMS) located
on a bus stop. There are three kinds of information that a transit passenger wants to
know: the pre-trip information, the en-route information and the in-vehicle information.
Pre-trip information is static information including schedules, maps, destinations,
fares, and policies. This information helps passengers in planning trips. It is updated
periodically for informing passengers about every change in schedule time of bus routes,
the new routes, or the new policies and fares, except the actual bus operational situations
such as delay, late or early buses. To know the scheduled arrivals, a passenger can use a
free brochure showing the bus map and schedule of the bus. Also, the passengers can
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5
access the transit agency’s website to find the schedules if they are provided. Calling
the transit representatives is also a method for passengers to know bus arrivals. By simply
providing the bus station number, a passenger can get the scheduled times via telephone.
From the passenger’s point of view, there is no certainty that the expected bus
will arrive on schedule. Hence, if transit agencies cannot provide passengers actual
arrivals or in other words, the en-route information, the waiting passengers can only
depend on their own waiting time experiences. Therefore, they will tend to arrive to the
stop a few minutes earlier than the schedule to avoid missing the bus. If the bus arrives
late, this increases the passengers’ waiting time. Even if the bus arrives on time, the
waiting time could be perceived to be long. On the other hand, when the bus comes to the
stop earlier than scheduled, passengers can miss the bus even though they come to the
stop on time. In this case, the waiting time is now not a few minutes but as long as an
headway. Moreover, if passengers do not know that they missed the last bus, they may
experience anxiety. For on-board passengers, in-vehicle arrival information to the
destinations can help them to have smooth connections with other inter-modal transit
services.
Obviously, the uncertainty of bus arrival time increases passenger’s waiting time
and anxiety, deteriorates schedule adherence, and reduces the smooth transition among
inter-modal transfer. Thus, passenger dissatisfaction is unavoidable if transit agencies still
provide their customers only the static information. To satisfy customers, transit agencies
should provide their passengers not only static information (e.g., pre-trip information) but
also en-route and in-vehicle arrival information. Interestingly, some transit agencies that
provide real-time information to passengers recognized that diversion to private
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6
automobile was low even when bus arrivals varied but passengers knew this information
ahead of time (Schweiger et al., 2003). Therefore, providing passengers real-time arrival
information is necessary and this will encourage passengers to use the transit system.
1.3 Problem Statement
Surveys carried out by some transit agencies revealed that the most important
factor influencing bus riders’ decision to ride a bus, surprisingly, is not the bus on-time
performance but the real-time information about the bus of interest. On-time performance
only comes as the second important factor. This shows that real-time information is
playing an important role in customer’s perception of transit service quality (Zhong et al.,
2003).
Passengers’ demand for real-time information on bus arrivals is pressing transit
agencies to meet that demand in order to increase or at least, to keep their customers.
Having recognized the importance of this task, various agencies in European countries, in
North America (i.e., the United States, Canada), and Asian countries (e.g., Japan, Taiwan,
and Korea) have been installing AVL-APC systems to assist their bus fleets as well as to
develop a suitable RETBAIS. However, most of these agencies are using AVL-APC data
mainly for off-line performance studies. This practice continues in spite of the fact that
much more is to be gained in on-line applications such as predicting bus arrivals,
predicting passengers boarding or alighting at stops, and proactive dispatch programs.
The use of APC data and AVL data is also different. While APC data downloaded
daily or weekly from on-board microcomputers have been used successfully for years in
the off-line analyses, for instance, passenger loads, on-time performances, etc., the AVL
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7
data have not yet been used adequately for neither off-line nor on-line services, except for
incident response by many agencies. The reason behind the failure in using useful AVL
data is that the transit agencies did not recognize the potential benefits achievable from
useful AVL data. Therefore, their AVL systems were not used to match and track
vehicles, and AVL and the APC systems have been used as stand-alone systems. This
practice limits the on-line applications based on archived AVL-APC data (Furth et al.,
2003, p.6).
The purpose of this research is to develop a methodological framework and the
constituent model for advancing dynamic predictions and thus to support the real-time
bus arrival information system. By using the AVL-APC data intensively, the model will
be suitable for transit agencies whose bus fleets are equipped with AVL and APC
systems.
1.4 Goals and Objectives
This research is intended to enhance the interests of transit passengers as well as
transit service providers. The study aims at increasing the quality of actual arrival
information (e.g., en-route and in-vehicle information) to be provided to bus transit
passengers. Also, this research is intended to support bus transit providers in term of real
time control and management. For achieving the first goal, a real-time prediction model
for bus arrivals was developed. To attain the second goal, the applications of the model
were further developed for bus bunching and punctuality prediction in order to help
dispatcher in managing the fleet.
Specific objectives of this research are:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8
1. To define a research framework and constituent modules for the development
of methods to support bus real-time arrival prediction.
2. To develop modules and integrate the modules to create a dynamic model that
can predict real-time bus arrivals and dwell times at every stop.
3. To evaluate the accuracy of each module and the integrated set of modules.
4. To develop bunching and punctuality prediction algorithms that can help bus
dispatchers in controlling the fleet under real-time situations.
5. To propose the architecture of a-real-time bus arrival system based on the
developed model.
1.5 Study Methodology
The overall research framework is shown in Figure 1.1.
The problem definition step covered the general lack of real-time information on
bus arrivals and dwell time. As noted in the previous section, the users of bus transit are
placing much importance on this type of information. However, for a number of reasons,
a very high proportion of public transit agencies do not provide such information.
Although a number of transit agencies have taken steps to implement AVL and APC
systems, they have not taken the next step to make use of the data to develop a
RETBAIS. Also in this step, the issues of the existing bus arrival prediction algorithms
were noted. Consequently, the problem definition step dealt with the characterization of
the problem to be solved. Here, broad research goals and specific objectives were
defined.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9
PROBLEM DEFINITION REVIEW OF ISSUES IN THE DEVELOPMENT OF A REAL-TIME BUS ARRIVAL INFORMATION SYSTEM
> EXISTING PROBLEMS IN USING APC & AVL DATA. > EXISTING PREDICTION ALGORITHMS: STRENGHTS & WEAKNESS
> RESEARCH GOALS AND OBJECTIVES > DEFINITION OF VARIABLES > DATA AVAIBILITY
DATA GENERATION DATA COLLECTION
VISSIM'S PARAMETER > BUS ROUTE SELECTION SELECTION > NETWORK & BUS ROUTE DATA
BUS TRANSIT NETWORK IN VISSIM 4.10 > APC & AVL DATA > DATA ANALYSIS
BUS OPERATIONAL SCENARIOS MODEL DEVELOPMENT. TESTING AND APPLICATION
VISSIM SIMULATION & DATA > THEORETICAL BASES OF GENERATION STATISTICAL PATTERN RECOGNITION AND NON LINEAR REGRESSIONS > MODEL' S MODULES GENERATED DATA ON BUS TRIP TIME AND ARRIVALS
> MODULE DEVELOPMENT DATA CALIBRATION PROCESS > MODULE COMBINATIONS
WELL CALIBRATED DATA ON BUS > MODEL' S REALIBILITY WITH TRIP TIME AND ARRIVALS SEVERAL SIMULATED BUS OPERATIONAL SENARIOS (CONGESTION,LANE CLOSURE INCIDENT)
> MODEL' S ACCURACY WITH ACTUAL DATA > COMPARISION WITH AN ADVANCED MODEL
> BUNCHING AND PUNTUALITY FORECAST
Figure 1.1: Research Framework
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10
In order to fulfill the research goals and objectives, further steps were
accomplished. Two next steps were the data collection and the data generation. These are
carried out side-by-side.
In the data collection step, the APC and AVL data collected from the OC
Transpo, accompanied with other data of the transit network, were used for two purposes.
On one hand, these data were mined directly for the model development and testing steps.
On the other hand, they were used as the inputs for the data generation step.
The purpose of data generation step was to generate data, given the assumptions
of different bus operational scenarios which are difficult to study in the form of field
experiments (e.g., congestion, incidents, etc.). The VISSIM 4.10 traffic software was
used for these simulation purposes. In order to make the simulated bus operations in the
transit network as similar to the actual operations as possible, a calibration procedure was
developed. After the calibration procedure was completed, the generated data were then
used for testing the model under each scenario.
The next step involved model development and model testing procedures where
the variables were defined and theoretical aspects were researched. Based on the AVL
and APC data and the use of statistical pattern recognition technique and nonlinear
regression methodologies, a model consisting of three modules and four sub-modules was
developed. This developed model has the capability to make real-time bus arrival time
and dwell time predictions. After the model was tested in terms of accuracy and
reliability by using simulated data obtained from computer simulations of several bus
operational scenarios, the model was also tested with real-world data. At the end of this
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11
step, the developed model’s prediction performance on bus running time, bus dwell time
and passenger activities were assessed.
Finally, for bunching detection, punctuality study, and dispatcher help, the
developed model was expanded by using predicted bus running times and dwell times.
1.6 Thesis Document and Organization
The dissertation consists of eight chapters.
Chapter 1 provides an introduction to the subject of this research, research
objectives, and the research study framework.
Chapter 2 is devoted to the literature review including current applications and
the potential prediction algorithms for bus running time and dwell time.
Chapter 3 presents a broad view of real-time bus arrival prediction model
developed in this thesis research. This model includes three linked modules: Running
Time Prediction Module (RTM), Dwell Time Prediction Module (DTM), Punctuality and
Bunching Prediction Module (PBM). The DTM is composed of four sub-modules,
including: Real-Time Boarding Passenger Prediction (RBSM), Real-time Alighting
Passenger Prediction (RASM), Regression sub-module (RESM), and Busiest Door
Prediction (BDSM).
Chapter 4 covers the first module, namely the RTM. In this chapter, a pattern
recognition process based on a non-parametric statistical technique is presented which
can predict running times on every downstream link of an en-route bus. The module’s
refined procedures are also presented in this chapter.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12
Chapter 5 commences with some discussions on previous dwell time prediction
algorithms, which have been described in the literature. Next, a framework for dwell
time module and its four sub-modules are presented, followed by the solutions and
procedures for the first two sub-modules. The text describes how several deficiencies of
previous dwell time prediction models reported by various studies are overcome. This
chapter also covers data collected by the OC Transpo on dwell time and the number of
boarders, alighters and other variables such as on-time performance, bus location, bus
types, and time of day. Several statistical assumptions on their relationships are examined
in order to find the best statistical model for estimating dwell time.
Chapter 6 presents the method for the estimation of the developed model. First,
the procedure to simulate several bus operational scenarios by using VISSIM software is
introduced. Second, for comparison purposes, two reference models for bus arrival time
prediction are described. One of these models can be currently considered as an advanced
model and the other as the simplest model. Third, the developed model and the two
reference models are compared in term of bus running time, passenger activities, and bus
dwell time predictions given different bus operational scenarios and a variety of data. The
comparisons are based on the Tukey’s test procedure in order to provide sound
conclusions. By using the evaluation system, models’ accuracy and reliability are
assessed.
Chapter 7 covers the methodologies for the on-line detection of bus on-time
performance and bus bunching. For this purpose, a method to estimate real-time
prediction interval is developed given a user-defined level of significance. On the basis of
probability theory and the historical bus arrival time distribution, two methods are
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13
developed, one for on-line bus on-time performance evaluation and the other for on-line
bus bunching detection. These provide the probability that such events may occur.
Finally, the discussions on announcing of bus arrival times as well as spatial aspect and
details of the database for the developed model are discussed.
Chapter 8 presents conclusions and recommendations. The contributions of this
thesis research are highlighted. Recommendations are provided that include future
research on different aspects of data, mathematical algorithms, and some limitations of
the developed model.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2
LITERATURE REVIEW
2.1 Introduction
The application of AVL and APC systems in bus transit is becoming popular in
North America and Europe. This enables the agencies to develop a RETBAIS. Based on
the review of a number of publications (i.e., syntheses, reports, and articles), this chapter
examines the current practices, the experiences, the benefits and problems. Also, the
prediction algorithms of bus arrival time and dwell time used by transit agencies around
the world are reviewed.
2.2 Real-Time Bus Arrival Information System: Current State of Development
The deployment of the AVL and APC systems in many transit agencies has a long
history dating back to the early 1970s. However, this became widespread through the
United States and Europe only since the late 1980s and 1990s for the purpose of
increasing operational efficiency. After years of using AVL and APC systems, transit
agencies realized that these systems not only can increase the operational efficiency but
also provide passenger real-time information which is bringing substantial benefits to the
transit providers.
14
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15
The fundamental elements for deploying a RETBAIS are: the technical
characteristics of the underlying AVL system, the real-time prediction models, type of
media to distribute the real-time information, related costs of the system, customer and
the media reaction to the system, and institutional issues with the system (Schweiger et
al., 2003, p.2). In the field of interest, the literature review focuses only on the first three
elements. Each element is discussed in a corresponding part, namely: the AVL and APC
systems, arrival and dwell time prediction algorithms, and media and communication
systems.
2.2.1 AVL System and APC System
2.2.1.1 Automatic Vehicle Location System
Automatic Vehicle Location (AVL) system is the backbone technology for a
RETBAIS. The AVL system tracks the dynamic location of a remote bus and transmits
the data to the control center. The transmitted locations of the bus along with other data
such as speed, acceleration, or historical data collected from APC systems and other
diagnostic systems on the bus, are used to predict the new location of the bus. As a result,
the arrival time at a selected stop is predicted. Different agencies are using different
technologies that belong to the four AVL categories. These, in the increasing level of
advanced information technology, are presented below.
- Dead-Reckoning (DR)-based AVL system: This is the oldest and simplest
navigation technology. The devices frequently used are odometers and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16
compasses. Data recorded from instruments must be transmitted by radio to the
central control for processing, analysis, and display. In a DR-based AVL system,
a bus is equipped with a heading sensor known as an adaptation of a magnetic
compass, wheel odometers, an information processor, and radio equipment. The
readings of heading sensor and the readings of wheel odometers are processed
by the on-board information processor and converted into modulated waves that
are compatible with the bandwidth requirements, workable with both receivers
and antennas, and transmittable to the control center. At the control center, the
waves received from a bus then will be decoded for tracking and displaying the
location of the bus.
- Radio Navigation or LORAN C-based AVL system: The C configuration of
the Long Range Navigation (LORAN-C) system is the triangulation with three or
more land-based transmitting stations. The instruments of a bus using LORAN-C
are: a LORAN-C receiver, antenna, a data processor, and land mobile
transceivers. Location is delivered based on the distances from the bus to a pair of
fixed radio stations’ locations. No new LORAN C-based AVL system is
anticipated in the future (Okunieff, 1997).
- Signpost and Odometer (SO)-based AVL system: The system, known as
radio frequency signposts, or signpost for short (both active and passive) uses
radio beacon mounted on the top of the facilities. In a passive signpost system,
each beacon mounted on the pole or structure above the height of the bus has a
unique ID. The beacon sends a low power coded signal in order to save
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17
electricity. When a bus passes a signpost, the specialized receivers attached on the
bus receive and decode the transmission from the signpost to recognize the
signpost’s ID. The location of the bus then is determined based on the last beacon
ID the bus detected and the distances travelled. Figure 2.1 represents a passive
SO-based AVL system. In contrast with passive system, each bus in the active
system has a unique ID on a radio channel designated solely for automatic vehicle
locating use. The vehicle transmission is continuous and at very low power.
Therefore, when the bus crosses a signpost pole equipped with a receiver, its
location is relayed and transmitted by that signpost to the control center. This
alternative transmits information via underground wire system so the bus does not
require a wireless transfer of data to the control center. Details on SO-AVL
system can be found in Skomal (1981) and Okunieff (1997).
- Global Positioning System (GPS) - based AVL system: Development of work
on GPS commenced in 1973 for the sake of supporting the positioning
requirements of military operations. As of 2000, the full constellation of 24
satellites was launched to ensure that there will always be at least four satellites
at all sites on the globe. Each GPS satellite transmits a unique navigation signal
on two L-band frequencies, the LI frequency 1575.42 MHz and the L2 frequency
1227.60 MHz. Both bands carry the navigation message and Standard Positioning
System (SPS). A bus equipped with GPS antenna and GPS receiver connected to
an on-board microcomputer receives GPS signals, determines its location and
sends the location data to the control center via wireless communication system.
Figure 2.2 depicts a GPS-based AVL system for bus transit
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18
G m » OgwrwwMiwfeMlfrytunk.
Figure 2.1: Signpost- based Automatic Vehicle Location (Source: http:// www.calccit.org/ itsdecision ) Notes: 1- The location antenna mounted on the bus receives the signpost signals and transmits them to the on-board Vehicle Location Unit (VLU). The VLU then recognizes the signpost’s ID. 2- The location of the bus based on the signpost’s ID then is transmitted to the control center via radio system.
Figure 2.2: GPS-based Automatic Vehicle Location (Source: http:// www.calccit.org/itsdecision )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19
- Differential GPS (DGPS)-based AVL system: In order to enhance the
accuracy, DGPS requires that one of the receivers to be located as a base location.
From the known location of the base, the bias errors due to the estimated position
of the orbiting satellites are calculated. A bus on the DGPS based AVL system,
like other GPS-based AVL systems, has a GPS antenna, GPS receivers, and an
on-board microcomputer. The required additional unit is a radio link that receives
correction from differential reference station.
22 .1.2 Automatic Passenger Counting System
Automatic Passenger Counting System or APC system is an automatic means for
counting boarding and alighting passengers as well as the time that the bus is at a specific
location. Figure 2.3 depicts the components of an APC system mounted on a bus.
GPS Antenna Onboard Computer Rear door Front door sensor sensor
Odometer Signal Engine Signal Power from Battery
OROR
Infrared Radio modem modem
Figure 2.3: The APC system mounted on a bus (Source: http://www.infodev.ca )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The heart of an APC system is the on-board microcomputer connected to the
counting sensors attached on the door areas of a bus. Each time a passenger gets on or off
the bus, the sensors will record the passenger activities and store data in the on-board
microcomputer. There are two counting technologies applied in the APC system: infrared
beam sensors attached on the door areas, and treadle mats mounted at the step levels.
Based on the order of the broken beam in the case of infrared beam sensor or the order of
stepping on the mat in the case of treadle mats, the APC system can differentiate if a
passenger is getting off or alighting on the bus. The data stored in the onboard computer
can be transmitted manually with data collector. To transfer data automatically, infrared
or radio modems are needed. In case of using infrared devices, when the bus stops
(usually at the garage or bus stop), two infrared links, one on the bus and the second on
the bus stop or garage area, download data from onboard computer automatically. If the
bus is equipped with a radio modem, data stored on the on-board computer can be copied
automatically at the distances of up to 500 ft between the radio transceivers. Time to
download a full day data is only about 20-30 seconds (http://www.infodev.ca ).
The APC system can work as a stand-alone system using one of the four AVL
categories mentioned above to match passenger activities to predetermined bus stops.
Figure 2.4 illustrates the integration of the APC and AVL systems. Such system can
measure over 20 variables including passenger activities (boarding and alighting), bus
locations (GPS coordinates, direction, bus stop location), and operational activities
(speed, running time, arrival and departure time, dwell time, stop distance, access time,
idle time, passenger load, etc). Moreover, an APC system integrated with the AVL
system and the radio systems can provide real-time data for real-time dispatching and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21
arrival data dissemination purposes (John, 1999). Therefore, the APC system has great
potential as valuable source of data for either on-line or off-line applications in
RETBAIS.
W ^GPSSiyijtei
Radio AntMMa
Doors lift APC (Automatic Passenger Counter) Overhead Signs Odometer Signal Priority Emitters w 1 £
Figure 2.4: AVL and APC Systems on a Bus (Source: Furth et al., 2003)
2.2.1.3 Uses of Retrieved AVL-APC data in RETBAIS
Transit agencies use collected AVL and APC data for two main purposes: off-line
performance analyses and on-line applications. While many agencies have used the
collected data for off-line studies, a few agencies use these data for on-line applications,
such as providing passengers en-route and in-vehicle information. The degree of using
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22
these data differs from one agency to another. The APC data were usually used in off-line
estimates while the AVL data were used for incident response, safety, and recently for
real-time location tracking. However, many agencies failed in using the AVL system for
both on-line and off-line estimations (Furth et al., 2003). Very few transit agencies (for
instant: Calgary Transit, Canada) considered using AVL data for off-line analysis as an
objective. While the APC system can collect and store many valuable variables useful for
on-line analysis, they have not been used for the real-time applications because they were
not integrated with the AVL and radio systems. In contrast, the AVL system, which is
considered a more “advanced” technology in real-time application, is not programmed to
store data. Consequently, it has been used only as a means for incident response
(Furth et al., 2003). Ironically, many agencies did have AVL system but still relied on
manual check to measure running time and on-time performance whereas other agencies
that had APC systems but did not integrate them with AVL systems and radio system still
could not provide passengers real-time arrival information.
Some successes in North America, European countries, and Asian countries in
implementing real-time bus arrival system are listed in Table 2.1. Some typical
applications are detailed in terms of technology and scope.
NextBus System. United States
The Nextbus System is well-known, serving 28 agencies over 13 states in the
United States. Based on the GPS-based AVL system, satellites and wireless
communication, the Nextbus System is capable of predicting real-time arrivals for buses
at all bus stops. This information is updated every minute and provided to passengers in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23
various ways. At bus stops, real-time arrivals are announced via dynamic message signs
(DMSs).
Table 2.1: Transit Agencies with Real-time Bus Arrival Information System
Agency Location AVL system Description Website address Real-time arrivals via www, palm pilot, Nextbus GPS www.nextbus.com DMS, and internet capable cell phone. Cape cod, RTA, GPS Real-time bus’s location via Internet www.capcodtransit.org Dennis, MA Los Angeles, Real-time arrivals via DMS at Metro LADOT, Signpost www.ladottransit.com rapid bus stops. LACMTA San Louis Obispo GPS Real-time bus arrivals at major bus stop Transit, CA Real-time bus arrival via telephone and Denver RTD GPS www.rtd-denver.com mobile applications Busview and Mybus programs to Seattle, King Signpost provide real-time information and http://mybus.org County Metro others via Internet and mobile phone Ohio State Real-time arrivals via website and GPS University message sign at stops Real-time arrivals via DMSs and real Tri-Met GPS time schedule via website. http://www.trimet.org ( Portland, OR) Metro transit, www.halifax.ca/metrot GPS Real-time arrival via telephone Halifax, Canada ransit/gotime.html TranLink, Vancouver, GPS Real-time arrival on 98-B lines. www.tranlink.bc.ca Canada London, United Countdown system provides real-time httD://www.tfl. 20v.uk/ Signpost Kingdom bus arrivals at stops via message signs buses DGPS and PROMISE, provides real-time arrival Helsinki, Finland Signpost via internet and wireless terminals Magdeburg, PIEPSER, provides passengers the GPS Germany delay information that may occur. Brussels, Phoebus: Real-time arrivals at stop and www.stib.irisnet.be/FR GPS Belgium Internet /3600F.html SNCF -Real-time arrival via media, Paris, France GPS internet. Kaohsiung, Taiwan GPS Real-time arrival via Internet www.mybus.com.tw Taichung, GPS Real-time arrival via Internet www.mybus.com.tw Taiwan
The arrival information is also posted on the internet at http:// www. nextbus. com.
Passengers can access the website, select the bus route, direction and stop in order to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24
know the dynamic arrival information. Passengers with Palm Pilot and internet capable
cell phones can also get bus arrival information at anytime and anywhere.
- Countdown System, London. United Kingdom:
Transport for London or London Bus services Limited has a very large number of
buses, a total of 5700 buses. The information system, called Countdown, was first piloted
in 1992 in an endeavour to provide real-time arrivals for passengers. In 1996, a full
scaled AVL system and Countdown Program were approved with an approximately 27
million dollars of budget. To date, 2000 countdown signs have been installed, and 2000
others are expected to be installed by the following years, covering 25 percent of the
stops and serving over 60 percent of all passenger journeys. All 5700 buses in the system
were equipped with signpost-based AVL system. The buses are localized by signposts in
the 5000 signpost-network covering the service area. Each bus location is polled every 30
seconds. The predicted arrivals are disseminated to passengers via internet and DMSs.
The information in the forms of audio and visual are in the plan. The Countdown system
is well received by passengers (Source: http://www. tfl. gov, uk/buses)
- Tri- County Metropolitan Transportation District of Oreson (Tri-Met)
Tri-Met can be considered as the pioneer among over 30 agencies in the United
States for using collected data from APC and AVL systems for both on-line arrival
predictions and off-line performance studies. One of the keys to the success of Tri-Met
after years of using the systems was the development of the Bus Dispatching System
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25
(BDS). Another key was that the AVL-APC interface provided the local referencing of
the APC data. Hence, it increased the utilization of data resources for different purposes
(Furth et al., 2003). At present, Tri-Met has a real-time bus arrival system called Tri-Met
Transit Tracker system that provides real-time arrival information for selected bus routes.
Passengers can access the website at http://trimet. ore to get bus arrival information. The
Transit Tracker project is aimed at deploying initially 250 bus stops that announce real
time arrivals to passengers. The expected rate of implementation is 50 stops per year.
2.2.2 Bus Running time Prediction Algorithms
Different arrival prediction models and algorithms have been developed by
various transit agencies and universities. The scale and the scope of these models vary,
depending upon data needs. While some models solely use data collected from the GPS-
based AVL system, others use data collected from the APC systems. Several models
require more data than others and additional technologies are used to collect such data
(e.g., road conditions, traffic conditions). This subsection describes non-proprietary
algorithms and models that have been documented and published.
2.2.2.1 Blacksburg (Virginia) Prediction Algorithms
Four algorithms were developed by Lin and Zen, (1999) to predict bus arrivals.
According to the authors, first, the location of the buses and stops are tracked and relayed
by satellites and GPS-based AVL system installed on the buses. To match the locations
of a bus to the locations of the stops, especially on the loop routes where some stops have
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26
the same coordinates (e.g., longitude and latitude) but in different distances, a matching
module was proposed. Second, four different algorithms were developed based on
different data needs: GPS bus location data, GPS bus location data + bus schedule table,
GPS bus location + bus schedule table + delay, and GPS bus location + bus schedule
table + delay + time check point. In the first two algorithms, the running time of a bus is
predicted based on a simple interpolation from the bus schedule table, the scheduled
travel time between two major stops, the ratio of the distance between two major stops
and the distance between any link pair between these two stops (e.g., from the tracked
GPS bus location to a stop or link). Having observed that dwell time at time check stop
was the major contributor to schedule adherence, the authors developed the last two
algorithms to take delays and dwell time into consideration. Delays in these algorithms
are calculated given the assumption that the lateness or earliness at an upstream stop is
perceived by bus drivers. These delays may be diminished because drivers tend to adjust
the speed to maintain on-time arrivals at check stops.
2.2.22 Portland (Oregon) and King County Metro, Seattle (Washington) Prediction Algorithms
The models created by Dailey et al., (2001) and Cathey et al., (2003) have been
applied on Mybus System in King County Metro (Seattle), and Tri-Met (Portland). The
model includes three components: the Tracker, the Filter, and the Predictor. According to
Cathey et al., (2003), the mechanism behind this model can be summarized as follows.
- Tracker: When a request is received regarding the arrival time of a bus at a
bus stop, the actual location of that bus is relayed by GPS-based AVL
system. To match this location to a particular trip of a bus route, an
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27
elaborate tracking methodology is used. The core of this methodology
is that the original GPS data on bus locations (e.g., longitude, latitude,
and time) are converted into distances from these locations to an
underlying pattern, called distance into pattern. Therefore, the bus
locations are stored and updated easily. This eliminates the
ambiguities of differentiating bus directions when a vehicle is
operating back and forth on a block over the same roadway.
Filter: A Kalman filter is applied to estimate the bus dynamic state. The state
vector X includes three elements: distance into pattern, speed, and
acceleration. At time step (k-1), before the AVL system on the bus
sends the new data, the prior bus state at time step k is projected. This
estimation is based on the current state at time step (k-1) and a
transition matrix. When new AVL data are available at time step k,
the optimal estimation of the new state (or the posterior projected
state) of the bus is calculated based on the prior state and a correction.
The correction is an optimal weighting of values done by multiplying
a Kalman gain with the difference between the actual measured value
of distance into pattern and the best prediction of its value before it is
actually measured. These values then will be used to predict the next
state of the bus at time step k+1 and the procedure is applied until the
bus actually arrives at the stop.
Predictor: Locations of the bus are updated frequently at a predetermined
time interval such as at every time point or constant time intervals.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28
The arrival is predicted synchronously at each time interval when
the new AVL data of the bus become available.
2.22.3 Texas Transportation Institute Algorithms
The TransLink Lab at the Texas Transportation Institute developed two
algorithms to predict bus arrival: 1-minute zone algorithm, and distance-based model. In
the former algorithm, the bus route map is divided into small 1-minute zones based on the
historical travel time data. When a current location of bus is detected, the travel time
from that location to the predetermined destination is then calculated by counting the
number of zones between these pairs. In the later algorithm, the variation in running
speeds in various time period of a day (e.g., the class time), and the distance to
destination are modelled to reflex the bus travel time (Chien et al., 2004).
2 2 .2.4 New Jersey University Algorithm
Chen et al., (2004) created a model to predict bus arrival on the New Jersey
Transit bus route 39. Unlike the previous models discussed above, this model uses data
collected from APC system intensively. Having argued that the artificial neural network
(ANN) cannot be practically used in real-time bus arrival prediction due to long training
time, the authors developed a combination of an ANN and a Kalman filter. First, data
collected from APC data and other data sources (e.g., historical running times, distances
between stops, weather conditions) were used as the inputs to train the ANN. The training
back propagation algorithm and several hidden layers were examined. The well-trained
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29
ANN then was used for initial predictions of running times between stops, given the road
conditions and traffic conditions. To capture the real-time positions of the bus, a Kalman
filter-based method was also developed. The values predicted from ANN are used as the
initial states of the bus at the first stop. Each time the bus arrives at the next stop, the new
location, the actual running time, and the actual arrival time are relayed by AVL and
APC systems, and then a Kalman gain loop uses these data ( e.g., the most updated data)
for updating the new states of the bus. Therefore, the predictions are adjusted with new
state and updated data.
2.2.2.5 University of Toronto Algorithms
Farhan (2002), Shalaby and Farhan (2004), developed models to predict running
times and dwell times for bus route 5 in Toronto (Canada). Two Kalman filter
procedures, which were first proposed by Reihoundt et al., (1997) for predicting running
time and dwell time, have been developed in detail (Shalaby et al. 2004, p. 48). The
predictions are made at every timepoint. To predict running time of a current bus on a
link, the average historical running time (e.g., 3 days in the past) and the current running
time of the previous bus on that link, are used as the initial state of the current bus. Each
time the bus comes to a next stop, the state is updated by a Kalman gain and the
predictions are adjusted with the most updated data collected from the AVL and APC
systems. The same procedure is applied for predicting arrival rate of boarding passengers
at every stop. The predicted arrival rate is assumed to be a uniform distribution, so dwell
time is the product of that rate and the headway. The model outperformed in arrival
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30
prediction in comparison with other models (e.g., the ANN-based model and the average
based- model).
2.2.2.6 Artificial Neural Network (ANN)-based Algorithms
Several ANN- based models applied in the bus transit field can be found in Chien
et al., (2002) and Kalaputapu et al., (1995). Unlike the prediction models discussed so
far, which require only AVL data or combined AVL-APC data, these models need
different data sources collected from field measurements or simulated from simulators.
Chien et al., (2002) proposed two ANN-based models to predict arrivals: link-
based model, and stop-based model. In the first model, a segment bounded by two bus
stops was considered as a series of traversed links. One bus route may have several
segments. On each link, the following data were required: volume, speed, delay, average
link queue time, passenger demand at stops, and bus travel distance on a link. The ANNs
with the variable listed above were trained by the back propagation algorithm. During the
training step, several parameters have been examined such as number of hidden layers,
learning rate. If there are M links on a segment, ANN will be applied M times to predict
bus running time on each link. Therefore, the arrival time at stop i+1 is the sum of the
departure time at stop i and the total predicted travel times of all M links of that segment.
Unlike the first model, the second model integrated the traversed links into one link.
Therefore, each segment between a pair of two stops is a link. The data that were used to
train the ANN are integrated as the means and standard deviations of speeds, volumes,
and delays. This model is simpler than the first one because the trained ANN is used only
one time to predict running time on the link. However, it also required more variables
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31
than that of the link-based model to train the ANNs. There were 9 variables in this model:
distance between stops, mean and standard deviation of volumes, mean and standard
deviation of link speeds, mean and standard deviation of delays, number of intersections
traversed between a pair of stops, passenger demand at stop. On the basis of mean
absolute predictor error, both models showed good predictions when they were applied to
bus route 39, New Jersey Transit.
22 .2.1 Classical Statistical Regression Models
These models based on the classical statistic regressions vary from simple
multiple linear regressions to non-linear regressions. Besides the data that can be
collected by the APC or AVL systems (for example, the stop distances, average running
times on the links, the number of passengers getting on or off the bus, the average speeds
of bus on links), more data were required such as average delay on the link, traffic
volume, densities, the number of left turning vehicles, type of vehicle, traffic signal
timing, etc. The models treated bus running time or bus delay as dependent variable and
other variables as independent ones.
Alfa et al., (1988) proposed three statistical models to investigate the bus running
time for the transit network in the city of Winnipeg, Canada. Many variables were
collected and 11 of them were carefully selected in order to get the most influencing
variables. After removing the variables which had small correlations with the response
variable, only four variables were retained: number of bus stops, number of stop signs,
number of traffic lights, and length of the segment. These variables then were used to
developed three models, one is multiple linear regression and two others are non-linear
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32
regression models. The second model was suitable for extrapolation of bus running time
while the first model was preferable for interpolation.
Abdelfattah and Khan, (1998) developed six statistical models to capture the
variations in bus transit delays under several bus operational scenarios: normal road
condition (model 1), bus route in business district area (model 2), bus route including
high proportion of heavy vehicles (models 3, 4), and one-lane blocked route (models 5 &
6). The data for building these models were measured in the field (the OC Transpo bus
route 2, Ottawa, Canada) and the simulated data obtained from TRAF-NETSIM software.
Thirteen variables were considered as independent variables and bus delay was used as
the dependent variable. To retain the most influencing variable, several stepwise
regression procedures in the SPSS software were applied. The variables with the highest
influence on bus delays were: densities of left turn and through vehicles, link length, bus
efficiency ratio, number of stations, number of stops, and heavy vehicle traffic density
per link. The first model was linear multiple regression type and the others were all non
linear models.
Patnaik et al., (2004) proposed a statistical regression based model using APC
data. The APC database was split into two parts: 80 percent database was used for
developing multiple regression models and the remaining for testing the developed
models. Eleven variables collected from the APC system were used as the inputs and two
models were developed using the stepwise regression method.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33
2.2.2.8 Other Prediction Models
Other prediction models applied for bus arrival were found in the literature and discussed briefly as follows:
Simulation model. Chien et al., (2000) proposed a model to predict bus
arrival based on simulation. Although the enhancements developed by
the authors for CORSIM (CORridor SIMulation) software were intended
to increase the capability of CORSIM to capture the waiting passenger,
these also were used to estimate stop-to-stop travel time. Based on the
proposed model, bus schedule of each bus route was specified by users
and used as input. Then the simulator generated the buses on the network
based on input schedule. Therefore, given simulated road conditions,
passengers, and actual bus schedule, the enhanced CORSIM could be
used to predict arrival times.
Non-real-time AVL data model. Having recognized that historical AVL
data are capable of providing estimated bus running time, Horbury
(1999) proposed some models to estimate moving time, bus speed and
dwell time. The signpost-based AVL system data collected from route 18
in London (England) were decomposed into moving time and dwell time
at stop. Moving time for a trip from the origin to the destination was
calculated by subtracting the total travel time and the dwell times at
stops. To calculate the dwell time, boarding and alighting time models
were constructed from the survey data. These models have simple linear
regression forms. As a result, the moving times can be obtained. Because
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34
the models only used the AVL historical data, it is suitable for off-line
estimations.
Bus stop-to-stop line model: Son et al., (2004) developed a real-time
prediction model to project both the travel times from a bus stop to a
stop line at an intersection, and from stop line to a downstream bus stop.
The arrival time at the first stop and the location of the bus were
recorded by the AVL system mounted on the bus ITS devices. Two
Kalman filters were used to predict these travel times. The predicted
arrival time at the second stop is the sum of the two predicted travel
times plus the waiting times at the signal.
GPS map-based model: Weigang et al., (2004) developed models for
estimating bus arrival time and bus speeds based on the GIS map. The
mechanism behind the models is that the bus route is divided into a
number of short, straight line sub-routes. The location of bus relayed by
GPS system then is matched to find the bus position in the GIS map and
an algorithm was developed to make an accurate prediction for the
distance from the tracked bus position. If bus speed information is
available, then the running time is simply projected by dividing the
distance by the speed.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35
2.2.3 Bus Dwell time Prediction Models
Most of the models developed in the past considered dwell time as a delay and
explored the influencing factors on dwell time. Prior to the applications of the APC
system to bus transit, because of high cost and labor-power in data collection, studies at
that time focused mostly on the relationship between dwell time and the number of
passengers boarding and alighting which were counted manually at point checks.
Back in 1974, Kraft and Berger found that the time for serving one boarding
passenger is about 4.5 seconds and dwell time for boarding is 2 seconds plus time for
serving M boarding passengers (Kraft et al., 1974).
Having studied dwell time and passenger activities in Lafayette, Indiana (USA),
Guenther et al., (1983) generated some conclusions about dwell time:
The number of passengers boarding and alighting at each stop follows
Poisson distribution only when they are relatively low.
Negative binomial distribution is acceptable as a descriptor of boarding
and alighting passengers under all volume conditions.
The assumption that dwell time per passenger is independent of the
number of passengers boarding and alighting may be erroneous.
The relationship between average dwell time per passenger and the total
number of boarding and alighting passengers at a bus stop was found as
£ = 5.0-1.2[ln(z)] (2.1)
Where: z is the number of boarders and alighters at a stop
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36
Another model developed for bus dwell time estimation can be found in the works
of Senevirate (1988). The author found that the Gamma distribution fits data of boarding
and alighting time distribution better than the Normal and the log-normal distributions
did.
In the early 1990s, the equations relating dwell time to the number of passengers
as well as other data collected by the APC and AVL systems such as, fare box type,
number of vehicle’s doors, lifting operations, bus schedule adherence performances, bus
locations, and bus speed were first developed.
Marshall et al., (1990) developed six regression models (four linear and two
exponential regressions) to relate dwell time to the number of passengers, the fare
collection methods (coins, bill) and the bus induced-delay. The authors concluded that
exponential regressions were suitable and did not create negative dwell time when no
boarding and alighting passenger are recorded.
Levine et al., (1994) considered dwell time to consist of five components
(boarding and alighting time, waiting time for other passenger activities, waiting time by
buses that advanced ahead of schedule, waiting time for re-entry to traffic stream, and
time for attending equipment). The Levine’s models were mainly devoted to low-floor-
bus boarding time. The findings showed that a low- floor bus saves on the average 17.5
sec per full bus per stop, 0.16 sec per passenger with unimpeded movement and 5.78 sec
per passenger using mobility aids.
The Highway Capacity Manual (HCM) 2000 suggested a formula for the
calculation of dwell time as follows
D = (Pa) ( 0 + (Pb)(h) + top (2.2)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37
where: D = dwell time in seconds
Pa = Number of alighting passengers through the busiest door
ta = Passenger alighting time per passenger
Pb = Number of boarding passengers through the busiest door
tb = Passenger boarding time per passenger
top = Door opening and closing time
Rajbhandari et al., (2003) used statistical regressions to generate four models for
dwell time as a function of the number of boarding and alighting passengers, and the
number of standees. Like other authors, Rajbhandari found that the dwell time has a non
linear relation with the number of passengers as shown below
DT= a +b (Total) +c (Total) (S) (2.3)
where: a, b, c = are regression parameters
Total = Number of boarding and alighting passengers
S = Number of standees
DT= Dwell time.
Dueker et al., (2004) collected data by using the APC system in TriMet transit
system for developing multiple regression models. The determinants of dwell time
include passenger activity, lift operation and other effects.
Bertini and Geneidy, (2004) also used data from TriMet to model the dwell time.
The data were 452 stop samples of 14 trips of bus route 14. The authors found that dwell
time and passenger activities are related as follows:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38
Dwell time (s) = 5.8 +0.85 Na +3.6 Nb (2.4)
Where: Na, Nb is the number of boarding and alighting passengers, respectively
Shalaby et al., (2004) proposed a Kalman filter to predict the passenger arrival
rate based on the historical data of the number of passengers arriving at a stop. Once the
passenger arrival rate has been predicted, it is assumed to be constant during the
headway. Thus, the predicted number of boarding passengers is the product of the
headway and the predicted passenger arrival rate. Dwell time is then calculated by
multiplying the predicted boarding passengers and an average serving time per boarder.
Fang and Min (2005) generated a model to estimate the dwell time. First, the
author surveyed the distribution of passengers boarding and alighting times via front door
and the rear doors. Second, they developed several probabilistic distributions (e.g., log
normal, Gamma, and Weibull) to fit the frequency distribution. The probability of the
service time duration for both alighting and boarding is predicted by using Monte-Carlo
simulation.
2.2.4 Media and Communication System
Media and communication system play an important role in the success of a
RETBAIS. Real-time arrivals will be useless if they cannot be disseminated immediately
to passengers. A bus fleet well-equipped with APC and AVL systems will not work
efficiently if the data collected cannot be transmitted sufficiently and dynamically to the
control center. This subject is discussed in two parts: information dissemination media
for transit users and communication system for transit providers.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39
2.2.4.1 Information Dissemination Media
Printed medium is the simplest way transit agencies provide information to
passengers. Besides this traditional method, passengers can access the telephone system
to get traveller information. The innovations in information technologies now enable
transit agencies to have various media to distribute information to passengers. Currently,
passengers can get transit information from cell phones, PDAs, pagers, Palm Pilots,
internet capable mobile phones, kiosks, video monitors, DMSs, and the Internet.
The DMSs and video monitors are prevalent at bus stops. The DMSs and Video
monitors are non-interactive displays where passengers can get only en-route or in-
vehicle information if such devices are installed in the buses. The information shown on
the electronic signs usually includes: time and date, route number and destination,
waiting time (countdown or time range), and service interruption. Kiosk, internet capable
cell phones, and the Internet allow passengers to get information actively. Therefore,
passengers can get all kind of information they need (pre-trip information, en-route and
in-vehicle information).
Providing information via Palm pilots, pagers, PDAs and cell phones is a
preferable method that transit agencies aim at because such devices are personal and
wireless. Therefore, they enable passengers to get real-time information anytime and
anywhere at low cost.
Some current deployments of dissemination media are listed in Table 2.1.
(Further details can be seen in TCRP Report 92, 2003).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40
2.2A.2 Communication System
Most of agencies use radio system to transmit data. As the demand for real-time
information is increasing, the requirements for data transmission are on the rise. This can
cause “overcrowding of spectrum” placed on the available radio channels dedicated for
the AVL system. Therefore, the agencies are looking for the alternative solutions such as
using “spread spectrum”, narrowing radio communication band, renting commercial
radio service, using infrared or high capacity frequency link (John, 1999).
2.3 Summary
In many transit agencies, AVL and APC have long been used for increasing
operational efficiency. However, it is in recent years that the agencies have recognized
the benefits of applying AVL, APC and other advanced technologies for the provision of
passenger real-time information.
The use of collected APC data for off-line analysis has been the practice in transit
agencies, but the AVL system was not considered as an objective. Many agencies
installed AVL systems for their bus fleet, but cannot provide real-time arrival information
to passengers. They use the AVL system only for incident response while they use
manual check for on-time performance estimation and running time analysis. The APC
systems have been used for off-line studies only in spite of the fact that these can provide
valuable data for on-line applications.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41
Bus arrival prediction models reported in literature very seldom use the collected
APC data for real-time prediction while it is known that an integration of APC and AVL
system with radio systems can meet the real-time information requirements. Almost all of
the prediction models are based on the GPS-based AVL data and other diverse sources of
data.
None of the arrival prediction models based on AVL data predict dwell times and
include them in the total predicted trip time. This may cause inaccurate predictions of
arrival for downstream stops when the bus serves crowded upstream stops.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3
MECHANISM OF THE PROPOSED MODEL AND AVL-APC DATA COLLECTION
3.1 Introduction
Following the examination of the previous works involved in real-time bus
running time and dwell predictions, this chapter presents the framework of the
developed model for real-time bus arrival predictions and its building blocks. This
chapter is organized as follows: First, the components of a bus trip time are investigated
in order to emphasize the roles of running time and dwell time in bus trip time analyses
and predictions. Second, various modules are described and their integration into the
complete model is covered. Last, the study design including the process of APC and
AVL data collection and data descriptions are discussed.
3.2 The Components of Bus Trip Time and Influencing Factors
To date, many models and algorithms have been developed to investigate the
factors that influence bus running time and delay. The most influencing factors found in
studies are shown in Table 3.1. The identified factors influence different components of
the bus trip time. Therefore, in analyzing and estimating accurately the components of the
42
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43
bus trip time, causal factors should be studied, given that their effects will ultimately
influence actual bus operation on the route.
Table 3.1: Factors that influence Bus Running Time and Delay
Studies Factors Afdail et al., (1988) - Number of bus stops Abdelfattah and Khan (1998) - Link length - Number of traffic lights - Densities, left turning and through vehicles - Proportion of heavy vehicles in mixed-traffic stream Abkowitz and Engelstein (1984) - The number of passengers boarding or alighting at stops. Senevirante (1988) - Dwell time Patnaik et al., (2004) - Service characteristics McKnight et al., (2003) Traffic congestion
According to Maloney and Boyle (1999), a bus trip time includes the following
components: (1) actual moving times between stops; (2) dwell time at stops; (3) traffic
signal delay; (4) general traffic delays, and (5) recovery time at the end of each trip.
7% 5% H moving time (59%) ■ recovery time (7%) a deadhead time (9%) □ manuevering time (7%) 59% ■ Dwell time (13%) a signal delay (5%)
Figure 3.1: Bus Trip Time Components (Modifiedfrom Manoley et al., 1999)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44
Maloney and Boyle (1999) showed that actual moving time between stops is the
primary component accounting for overall 59 percent of total trip time, followed by
recovery time (13%), deadhead time (9%), time maneuvering to access and leave the stop
(7%), dwell time (7%), and signal delay (5%) (see Figure 3.1). The following paragraphs
briefly discuss each component of the bus trip time.
3.2.1 Actual Moving Time
Actual moving time of a bus is the time that the bus is in motion. It is heavily
influenced by three main factors: schedule, distance between stops, and speed of the bus.
A short schedule will force bus drivers to speed up whenever possible and rush to
the stops. Therefore, a short schedule may shorten the moving time of the bus in some
cases, but may raise safety concerns on the road and put pressure on the driver and the
dispatcher to keep the schedule adherence. In contrast, a long schedule may lead the bus
to come to the stops earlier than scheduled time. If the arrivals are consistently early, the
moving times may be increased intentionally by operators (i.e., to slow down the bus). As
a result, passengers will spend more time in underutilized buses (Senevirante, 1990).
The spaces between bus stops also influence moving time. A short distance does
not permit the attainable cruise speed of buses since the available distance between stops
is used for deceleration and acceleration operations. On the other hand, Levison (1983)
pointed out that the fewer the bus stops, the greater the number of passengers waiting at
downstream stop. Therefore, elimination of some stops will enhance average speed but
can increase access time for passengers. However, Levison found that on the basis of
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45
statistical analysis, reducing the number of stops did not influence the average speed to a
high degree given the fact of traffic signals.
The speed of the bus is an important factor that influences the bus movement
time. However, speed is affected by many factors. Traffic densities and congestions are
the major causes of low speeds of buses running in mixed traffic bus routes in the
downtown area. For instant, the kilometers travelled by a bus in one hour decreases
significantly from 22.5 km/h in suburban area to 16 km/h in the city, and 8 km/h in the
central business district (CBD) (Levinson, 1983). The speed of bus during the rush hour
periods was 1.4 to 1.6 times as slow as that of the car (Levinson, 1983) or equal to 42-59
percent of car’s speeds (McKnight et al., 1997). In a manner similar to the effect of the
number of bus stops, the number of traffic signals also reduces the speed of the buses
when they approach and leave the traffic signals.
3.2.2 Dwell Time
Dwell time is the time required for the bus at stop to serve at the busiest door plus
time for closing and opening door (HCM 2000). Major factors that affect dwell time
found by many studies are the number of boarding and alighting passengers, bus stop
spacing, fare collection method, type of bus, and type of on-board circulation. However,
the relationship between dwell time and the number of boarding and alighting passengers
was considered by many researchers to be sufficient for their purposes. Many models
have been developed to investigate this relationship. They vary from simple descriptive
statistical models (Bertini et al., 2004; Rajat et al, 2003; Lin and Wilson, 1992), to
probabilistic models such as Poisson model (Chien et al., 2000), Gamma model
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46
(Senevirante, 1988), natural logarithm model (Guenthner et al., 1983), and exponential
model (Leo et al., 1990). The previous works on dwell time shows another trend, namely
to project dwell time based on the relationships between dwell time and the bus service
characteristics. The models belonging to this trend can be found in following publications
(Kraft et a l, 1974; Kostopolous et al., 1985; Dueker et al., 2004). These models are the
multiple regressions where dwell time is the dependent variable and types of fare
collection, number of doors, and on time performance are used as independent variables.
3.2.3 Traffic Signal Delay
Traffic signal delay accounts for a large proportion (i.e., up to 32- 45 percent) of
total delay (Levison, 1986). Most of this delay is due to waiting and queuing at signalized
intersections. In the Highway Capacity Manual 2000 (TRB 2000), a bus in a traffic flow
is converted into equivalent passenger car units so that an average delay per queuing
vehicle is calculated. This value can be considered as the delay of a bus at a signalized
intersection.
3.2.4 General Delay
General delay refers to delays caused by traffic congestion, crowded bus berths,
and lane changing maneuvers. It is fair to say that bus travel time delay is not only caused
by ordinary vehicles sharing the lane with buses but also by the buses themselves. Slow
and large buses in heavy traffic stream, like other heavy vehicles, cause car drivers to
avoid travelling in the same lane and consequently result in intricate lane changing
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47
maneuvers. Hence, this characteristic of traffic flow increases delay. According to
Ubitran et al., (1986), traffic congestion has a severe adverse impact on bus travel time.
So far, very few studies have been found in the literature that examined the direct impacts
of congestion on bus travel time. Currently, MacKnigh et al., (2003) developed models to
quantify the impacts of congestion on bus travel time in their studies of 39 bus routes in
Northern New Jersey (USA). The following is such a model:
dBTTr ff = 0.73(77Tr fl -T T T r f f ) (3.1)
Where:
dBTTrjf ~ Change in travel time rate (minutes per mile) due to congestion for
buses on route
T T T r a = Travel time rate for traffic under existing conditions
T T T rj f = Travel time rate for traffic under free flow conditions
0.73 = Statistical parameter resulted from calibration study
To quickly estimate the actual travel time rate, travel time rate indices (TRIs),
which are the extra amounts of time (in terms of minutes/mile) for travel over a link,
were used.
TRIS = TTTr a / TTTrj f or TTTr a= TRIs. TTTrj j (3.2)
From Equations 3.1 & 3.2, bus travel time increment due to congestion is as below.
dBTTr ff = 0.737T7; ff(TRIs -1) (3.3)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48
3.2.5 Recovery Time
Recovery time is the required time for layover of a bus driver at the end of each
trip. The length of recovery time is based on the labor agreement with bus drivers and it
is usually equal to at least 10 percent of a round trip time (Pile et al., 1998).
Obviously, if buses are running on the exclusive bus route or on the Transitway,
many factors influencing the travel time components under mixed operations are
eliminated. In contrast, if buses are running in the mixed-traffic bus lanes where other
vehicles share the same lane with buses, the buses are vulnerable to delays caused by the
many frictions such as traffic volume, incidents, curb parking operations, etc. For this
reason, predicting the influences on the components of a bus trip in mixed traffic bus
route will continue to be a challenge. For information, it should be noted that mixed-
traffic bus operation accounts for almost 99 percent of total bus route distances in North
America (Kittelson and Associates Inc., 2003).
3.3 Proposed Model Structures and Components
3.3.1 General Discussion
As discussed above, bus travel time is complex and is influenced by many
stochastic factors. Hence, to predict bus travel time, one should consider all bus trip time
components. The recovery time component is an exception which does not influence the
accuracy of bus arrival time predictions. Other identified components play a role in the
success of a bus arrival prediction model. If any of these components is not adequately
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49
modelled, one may get poor predictions of bus arrivals, especially when the bus operates
in the mixed traffic route during rush hours.
The models reported in literature can be divided into three groups. In the first
group are the models which aggregate all components (e.g., actual moving time, general
delays, traffic signal delays, and dwell times) into one variable called running time and
predict its value. These models can be found in the works of Chen et al. (2004), Chien et
al. (2002), Abdelfattah et al. (1998), and Kalaputapu et al. (1995). The second group
includes the models that integrate moving time, general delays and either traffic signal or
dwell time into the running time. The bus arrival prediction is the sum of that running
time and estimated traffic signal delays or dwell time at stops. Please refer to the works
of Shalaby et al. (2004), Son et al. (2004), Lin and Zen (1999), and Horbury (1999) for
further details of these models. The last group includes the models that take into account
only one main component of bus travel time, the actual moving time. These models are
based only on the data of the actual location of the bus provided by the GPS-based AVL
system and make the prediction of a new location of the bus in the next time step. Some
of these models can be found in the publications of Dailey et al. (2001), Cathey et al.
(2003), and Weigang et al. (2004).
Each group has drawbacks and advantages. The models in the first group do not
require as elaborate data as required by the second group. However, because of the
aggregation, these models cannot capture the effects of one component on another. For
example, a long moving time can cause a crowded downstream stop(s) and this, in turn,
results in a long dwell time contributing to a late arrival at the following stops. On the
contrary, the models in the second group which separate dwell time as an independent
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50
component can differentiate the effects. Consequently, they are more flexible than the
first ones. Because of their flexibility, such models may be applicable for both arrival
time prediction and dynamic bus control actions. The models in the last group were
advanced for tracking the dynamic bus locations. However, based only on the locations
of the bus and ignoring dwell time and other components of a bus trip time, the accuracy
of these models is good for an arrival prediction of one stop ahead but it may deteriorate
at the downstream stops after that stop.
The model proposed in this study takes into account the advantages of different
models aforementioned. In order to use data collectable by APC and AVL systems,
moving times, traffic signal delays and general delays are integrated into running times
while dwell time is treated separately. The proposed model has three modules. The first
module is for predicting the running time, the second module is for predicting dwell time,
and the last module is for model application. By separating dwell time and running time
prediction, the proposed model can capture the effects of running time on dwell time and
vice versa. Also, basing on the latest updates of the dynamic locations of the bus, each
module will adjust its predictions at every time step. Therefore, the whole model can be
applied for real-time predictions.
3.3.2 Assumptions
The following assumptions are made in this research study:
- A bus route under consideration is composed of links and bus stops/time-
points where a bus link is a route section whose two ends are two bus stops or timepoints.
- All buses in the bus route are equipped with APC, AVL and radio systems.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51
- Each time a bus arrives at and departs from a stop, the following data are
recorded and transmitted dynamically to the control center: location of the bus at this
stop, the bus trip ID, the times the bus arrives and departs the stop, the number of
passengers boarding and alighting at the stop, and the running time the bus travelled on
the previous link.
Figure 3.2 presents a simplified sketch of bus operation on a route.
Stop3 Stop4
AT4 ■link Travel Tffne 3
AT3
AT2 JJnk Trawl Tine 1
AT1
STOP
Figure 3.2: A Simplified Sketch of Bus Operation on a route
3.3.3 Structure and the Building Modules of the Proposed Model
As mentioned above, to take advantages of the previous models, the proposed
model for bus arrival time prediction in this study is the combination of the three
modules. The first module predicts running times and is therefore called Running Time
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52
prediction Module (RTM). The second module predicts dwell times, and it is called
Dwell Time prediction Module (DTM). The last module predicts bus punctuality and
bunching, and it is called Punctuality and Bunching prediction Module (PBM).
Running times in a bus link are predicted by the first module (RTM) in the
following manner. Given a current running time situation, the historical data of running
times of each link are used as a basis to predict running times. Each time the bus comes
to a stop or when the data collected by APC and AYL become available, the most
updated data are used to predict new bus running times. These data are also stored to
update historical data. The running time prediction module is described in details in
chapter 4 and 6.
Dwell time and passenger occupancy prediction module (DTM) is based on the
passenger activities recorded at the stops (i.e., both historical data and the current data).
The proposed dwell time prediction module is presented in chapter 5 and 6.
The punctuality and bunching prediction module (PBM) is described in chapter 7.
In this module, a method to estimate prediction interval is to be examined. Based on this
method and by using probability theory, two algorithms are developed for bus bunching
and bus on-time detection.
In general, to predict bus arrival/departure time at stop i+1, given the recorded
departure time of the bus at previous stop (/) denoted as Dpactuai, /, the first two modules
( i.e. RTM and DTM) are connected together by two simple general equations:
Afi+i~ Dpactuau+ r,•_;•+/ (3.4)
Dpi+]= Ari+i + dwelli+i (3.5)
Where:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53
A r i+] is the predicted arrival time at stop /+/
Dpi+i is the predicted departure time at stop i+1
Dpactuai, i is the actual departure time of the bus at stop i
fij+j — predicted running time the bus will spend on the link between two
stops, obtained by RTM.
dwelli+i = predicted dwell time of the bus at stop i+1, obtained by DTM
Once the arrival prediction has been made for a bus stop, this value will be
compared to the scheduled arrival time that the bus is supposed to be at that stop. This
predicted time difference will indicate if the bus is likely to be early, on-time, or late.
Based on this information and other predicted variables, the PBM will provide the best
suggestions to bus dispatcher in support of decision making.
3.3.4 The Mechanism of The Proposed Model
Figure 3.3 shows the process of real-time prediction of the proposed model. For
detailed descriptions, let us assume that the bus route under consideration has n bus stops
denoted as S T O P i (where i=l to Yl) and bus k has just arrived at bus stop ST O P i. We call
the actual time that bus arrived at stop i as A r actuai_ , and the actual time the bus left the
stop as Dpactuai, /• It should be noted that the time A r actuan may be different from the
scheduled time Sct of bus k at stop i due to travel time variations. Let us denote A rj and
Dpj as the predicted arrival time and departure time of bus A: at a downstream stop j
(where j= i+1 to n).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54
START
HISTORICAL DATA
MODEL TRAINING PROCESS
RUNNING TIME DWELL TIME PREDICTION PREDICTION MODULE MODULE
SAME DAY HISTORICAL DATA RTM DTM
REAL-TIME APC & AVL DATA
Every h seconds
MODEL PREDICTION PROCESS
RTM PREDICTION DTM PREDICTION
REAL-TIME ARRIVAL PREDICTION AND PASSANGERS’ ACTIVITIES
MODEL APPLICATION
PUNCTUALITY & BUNCHING PREDICTION MODULE PBM
> PUNCTUALITY PREDICTION > BUNCHING PREDICTION
END Figure 3.3: Process of Real-time Prediction
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55
To predict the arrival times of bus k at downstream stops (e.g., i+J,
i+2.. .n), the following steps are applied.
1. Initial Arrival Predictions:
• Running time module will predict running times of bus k on all
downstream links (i.e., LINK ij+i, LINK i+ij+2 ...LINK ). The
predicted running times are coded as r , r i+jj+2 ... r
• Dwell time module will predict dwell times of bus k at all downstream
stops (i.e., STOP ,•+/, S T O P i+2,.., S T O P n). The predicted dwell times
are denoted as: dwell i+j, dwell i+2—, dwell n.
• Because the actual departure time at stop i was recorded ( D p a c t u a i, ;),
the predicted arrival of bus k at the next stop ( i+1) is
Ari+j Dpactuai, i + r i _ i + l (3.6)
• The predicted arrival time of bus k at stop i+2 then will be calculated
Ari+2 = Dpi+i + ri+ij+2 - (3.7)
Dpi+1 = Ari+ 1 + dwell i+/. (3-8)
Substitute Dpi+j of Equation 3.8 into Equation 3.7, so we have:
Ari+2= Ari+i + dwell ,-+/+ n+ij +2 (3.9)
Substitute A r ,+/ of Equation 3.6 into Equation 3.9, so we have:
Ari+2 Up actual, i + ri_i+I+ dwell i+l+ ri+l_i+2 (3.10)
• The same mathematical operation is applied for all other stops. For
example, the predicted arrival time of bus k at stop i+3 will be
calculated as:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56
Ari+3= Dpi+ 2 + ri+2j+3 (3.11)
Dpi+2 = Ari+2 + dwell i+2 (3.12)
Substitute Dpi+2 of Equation 3.12 into Equation 3.11. We have:
Ari+3= D p a c t u a i i + nj+i+ dwell ,•+/+ ri+ij+2 + dwell i+2 + ri+2j +3 (3.13)
Hence, we have initial predicted arrivals of bus k at all downstream stops
even when bus k is currently at an upstream stop.
2. Update Arrival Predictions
The predicted travel time to each down stream stop will be updated when
the most recent bus arrival information becomes available at every h seconds. For
example, as the bus departs from stop 1, the running time prediction module,
together with dwell time prediction module, predicts arrival times from first stop
to all downstream stops in the fashion as mentioned above. By the time the bus
arrives at stop 2, new updated arrival information, running times and passenger
activities of the buses on the bus route have already been collected by the AVL
and APC systems. These data then will be used by the two modules to make new
predictions of running times and dwell times. As a result, the arrival predictions
will be updated.
In general, when bus k comes to the next stop (e.g., stop i+1), the new
actual arrival time and departure time were recorded as Aractuai ,+/ and D p a c tu a i, /+/,
respectively. New updated arrival predictions then will be calculated at time step
D p actual, i+1 as follows:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57
• Running time module will use updated information to predict new
running times of bus k on all downstream links after stop i+i(e.g.,
LIN K i+ij+2, LINK i+2j+3 ...,LINK The new predicted running times
are coded as r (l> i+ij+2, r (l) i+2_,-+j .... r (,) The superscripts attached
with each variables indicates the number of times the predicted running
times have been updated. In this case, the running times have been
updated for the first time.
• Dwell time module will use the most updated information to predict dwell
times of bus k at all downstream stops (e.g., S TOPi+2,STOP i+3 ...,STOPn).
The new predicted dwell times are denoted as: dwell (I)i+2 , ...,dwelfl)n.
Like running time variables, the superscripts indicate the number of times
that dwell times have been updated.
• New updated arrival prediction at stop (i+2) will be calculated as follows
A r 0 )i+2 = D p actual,i+I+ r ( 1)i+ij+2 ( 3.14)
The same mathematical operation will be applied to update the arrival
predictions for all downstream stops. For example, the updated arrival time
of bus k at stop i+3 will be as follows:
A r (I)i+3 = D p (I)i+2 + r (1)i+2j +3 (3.15)
D p (1)i+2= A r (I)i+2+ dwelt l)i+2 (3.16)
Substituting Equations 3.14 and 3.16 into Equation 3.15 gives
A r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 • The same procedures will be applied every time the bus actually comes to a downstream stop (e.g., stop i+2, i+3..., n). For example, the prediction on arrival time of bus k at stop (i+3) is updated when bus k arrives at next stop (i.e. stop i+2) as shown below. At~ ^ ^ i+3 — Dpactuai, i+2 + ^ ^ i+2_i+3 (3.18) If we compare the first updated value of predicted arrival time at stop (i+3) (Eq. 3.17), and that of the initial prediction (Eq.3.13), we can see that the new value is updated based on the two most recent updated information. First, it is based on the actual departure time recorded at new stop D p actuaij+ i. Therefore, the model can capture the adverse effects of an incident that may cause a late departure at stop (i+1) when the bus was running in the LINK Second, it is based on the first time updated running times (i.e. / /yl i+ij+2 , r(,) i+2j +3 ) and dwell time dwelf^+ 2 . These variables are updated by the RTM and DTM, given the newest data collected from the APC and AVL systems. Once again, if we compare the first updated arrival time at stop (i+3) in Equation 3.17 and the second updated one in Equation 3.18, we can see the second updated arrival based on the new actual departure time at stop D p actuai,i+2 and the second updated running time r<2) i+2j+3- Hence, the model can capture the dynamic changes in bus travel times. 3. Interpolate Arrival times at Skipped Stops Following the mechanism presented above we can see that, at each time step, or in other words, at each time the bus left an upstream stop, say Dpactuan , the predictions of arrivals of this bus at all following stops will be made based on the actual recorded time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 Dp actuals, the predicted running times and the predicted dwell times dwellj (j= i+1 to n). A question can be asked as to how the proposed model predicts the arrivals at downstream stops when the bus comes to an upstream stop without stopping due to serving requirements? In practice, if the bus does not stop at a bus stop, neither location of the bus stop nor passenger activities at this stop are recorded. In this case, dwell time is definitely zero but the arrival time is not known. In case of a bus equipped with GPS-based AVL system, the interpolated time that the bus approaches the stop can be considered as the GPS time recorded when the bus was closest to this stop. However, if the bus did not have a GPS-based AVL system, (e.g., the bus has another AVL technology such as S/O-based AVL or DR-based AVL), the arrival time of the bus can be estimated as follows: - Assume three bus stops A, B, and C and that bus k passed stop B without stopping. - The previous buses had the running times in LINK a b are r(k-d) (where d= 1,2 ..). Approximately, the interpolated arrival of bus k will be Ar(k) interpolated, B Dp (k) actual, A + f (k) Average A_B ( 3 * 1 9 ) Where: r(k) Average a_b is the average of the same-day historical running times of the previous buses running on the LINK a _ b in one hour period. 3.4 APC and AVL Data Collection Based on how the proposed model works (i.e. its mechanism) and its building elements, a study design containing the processes of data collection and data reduction was developed. It includes: data collection, data description, and data reduction. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 3.4.1 Bus Route Selection In order to develop a model capable of predicting real-time bus arrival under different operational scenarios such as buses operating during off-peak and peak-hour, on mixed traffic route or on exclusive bus lane, the selected bus routes have to cover these situations. In this study, data retrieved from APC and AVL systems of the OC Transpo were used. Two bus routes were selected as the study subjects; route 95 (Orleans-Fallow Field) and bus route 1 (South Keys-Ottawa Rockcliffe). The reasons behind the selections are: - Bus route 95 is one of the three major bus routes in Ottawa (i.e. route 95, route 96 and route 97) and it operates 22 hours a day from 4:00 a.m. to 2:30 a.m. Almost all of the buses (95%) operating on this route are articulated buses. - Almost all of the segments of route 95 are on the Transitway and the others are designated as exclusive bus lanes located on highway 417, Albert and Salter streets. - Route 1 is a mixed-traffic bus route located almost entirely on Bank Street, an important radial road in the city of Ottawa. - Both bus routes cross the central business district (CBD) of Ottawa. Bus route 95 has 63 stops. It connects Orleans to South Nepean and is 31.4 kilometers in length. In this study, the route segment shown in Figure 3.4A from Blair to Lebreton station was selected because it contains both Transitway and exclusive bus lanes. This segment is 10.26 km long; it includes 26 stops and the average distance between two stops is 0.39 km. Out of 26 stops in this segment, seven stops were chosen and their data are collected and tabulated in Table 3.2. The schedule headways during Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 weekday vary from 3 to 5 minutes in peak-hour to 15 minutes after 10 p.m., and 30 minutes after midnight. Iasi LEBRETON Route 95 BLAIR-LEBRETON 1127128 ST. LAURENT TRAIN BLAIR CYRVILLE Figure 3.4A: Bus Route 95 (Blair-Lebereton) Table 3.2: Distance between stops BLAIR and LEBRETON. Bus Route 95 (Orleans-Nepean South) Stop ID STOP DESCRIPTION Distance between stops Accumulated (km) Distances (km) EE915 BLAIR 2B 0.00 0.00 EB 905 ST LAURENT 2B 2.79 2.79 AE900 TRAIN 2A 1.19 3.98 CE940 LEES 2B 2.16 6.14 CD910 MACKEZINE KING 2A* 1.87 8.01 CA920 ALBERT-KENT* 1.19 9.2 C J 900 LEBRETON* 1.06 10.26 Note: * bus stop is in CBD area. Please see Appendix A for the map of route 95 Bus route 1 has 65 bus stops lying between Ottawa Rockliffe and South Keys. It is a mixed-traffic bus route because it is designated mainly on Bank, Wellington, Rideau, St. Patrick and Beachwood streets. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 Route 1 SOUTH KEYS-RIDEAU Figure 3.4B: Bus route 1 (South Keys-Rideau) Table 3.3: Distance between stops GREENSBORO and RIDEAU. Bus Route 1 (South Keys - Ottawa Rockliffe) Stop ID STOP Distance between Accumulated DESCRIPTION stops (km) Distances (km) RF900 GREENSBORO 1A 0.00 0.00 RF075 BANK-JOHNSTON NS 3.19 3.19 RA190 BANK- WALKLEY 1.19 4.38 RA945 BILLINGS BRIDGE 4C 2.70 7.08 CH080 BANK -HOLMWOOD* 2.12 9.20 CF630 BANK- GLADSTONE* 1.55 10.75 CD 920 RIDEAU 3A* 2.1 12.85 Note: * bus stop is in CBD area. Please see Appendix A for the map of route 1 In this study, the segment connecting Greensboro stop and Rideau stop as shown in Figure 3.4B, is selected and seven bus stops were considered as timepoints (see Table 3.3). The schedule of bus route 1 varies depending upon the time of day, from 12 minutes in peak-hour period to 30 minutes after 7 p.m. 3.4.2 Data Sets Data from the APC-AVL systems available from the OC Transpo for each selected stop were clustered by time periods in a day and seasons in a year. In this study, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 only weekday data were of interest. Weekday peak period data are considered as the ones that record bus arrivals at stops from 8 a.m. to 10 a.m., 12:00 noon to 2 p.m., from 4 p.m. to 6 p.m. in the afternoon. Weekday evening is considered as the time of bus arrivals after 7 p.m. However, this study did not consider weekday evening. Data were also clustered by season in order to capture the various passenger activities and on-time bus performance in different seasons. Four seasonal data sets, namely SEASON1 (January 1st to April 30th), SEASON2 (May 1st to 31st Jun), SEASON3 (July 1st to August 30th ), and SEASON4 (September 1st to December 30th ) were also clustered. The data for bus route 95 and bus route 1 were collected for one year, covering from Season 4 of 2004 to Season 4 of 2005. The number of observations during peak-periods in each sample is shown in Table 3.4. Each observation noted in Table 3.4 contains a set of information, including stop ID, route and route direction, time period and date, actual and schedule arrival, the number of passengers boarding and alighting at a stop, the number of passengers on-board when the bus leaves the stop or passes it. These data are synthesized in the tabular form called APC point check as shown in Figure 3.5. Table 3.4: Data Sets for Bus Route 95 and Bus Route 1 SEASON 1 SEASON[2 SEASON[3 SEASON 4 Route 8-10 12-2 4- 8-10 12-2 4- 8-10 12-2 4- 8-10 12-2 4- a.m. p.m 6p.m. a.m. p.m. 6p.m. a.m. p.m. 6p.m a.m. p.m. 6p.m. 95 472 998 1220 890 744 931 849 494 700 592 450 600 1 513 335 189 291 203 113 297 210 130 389 312 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 PROCESSED: 20G5-Q9-27 1 3 :5 6 :1 8 AUTOMATIC PASSENGER COUWTIHG STSTBt POINT CH2CX WEEKDAY SERVICE B0GKIK3; SEP04 STOP: CC19Q OLD ST PATRICK C08QWG HS PERIOD: 2384-09-05 TO 2804-12-13 ROUTE: 1 SOUTH KBYS-DOSfflTCWCEHTRKVlI. TIME: 1 6 :0 0 :0 0 TO 1 8 :8 0 :0 0 BISECTION: I H83D DAY OP THE WEEK: M3S TOE WED THU FBI LOAD AT ACTUAL DIFFERENCE OKS QF7S DEPARTURE DATS 16:89 1 0 1 1 2004-11-82 TUE 16:19 11 0 2 0 2004-11-17 WED 1 6 :21 13 0 I 0 2QD4-11-19 FBI 16:15 7 0 1 B 2004-11-23 TOE 16:14 6 0 1 D 2004-11-25 THU AVERAGE 9 0 1 0 (5) Figure 3.5: Structure of APC Data at pointcheck 3.4.3 APC and AVL Data Description The data sets used in this study were collected from the APC system mounted on the buses. The OC Transpo has been equipping APC system on about 10 percent of their bus fleet. These units are circulated among all bus routes routinely to get the operational data of all bus routes for analysis purposes. At present, the location of a bus is identified by the odometers and signpost-based AVL system. The GPS-based AVL system has been deployed in the OC Transpo and was expected to be complete at the end of 2005. The APC system can collect many data items automatically, and at the same time. Table 3.5 presents the descriptions of data collectable by APC and AVL systems of the OC Transpo. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 Table 3.5: Data Collectable by APC and AVL systems Collectable Data Description SCHED DEP TIME Scheduled departure time at specified section DATE The date on which the observation occurs TIME OF DAY Time period o f bus operation DOOR OPENING TIME Bus door opening time DOOR CLOSING TIME Bus door closing time EXCESS_ TIME The portion of time spent at a stop other than servicing passengers; that is , the stop time at a stop excluding the dwell time ATC DEP TIME Actual departure time MOVING_ TIME Time spent Moving Between Stops; The proportion o f time between departing the first stop and the arrival at the second stop-excluding any idle times, stop times, and stop- and- go times. STOPGO_ TIME The portion o f the running time spent in a stop and go situation (stop time less than 10 seconds). IDLE TIME The portion o f the running time spent in an idle situation. DWELL TIME The portion o f running time spent in servicing passengers EXPECTED TIME The scheduled time at the stop, as obtained from the itinerary ACTUAL TIME The actual time that observed trip passed the stop. ONS I, ONS 2, ONS 3 The number o f boarding passengers at the bus stop used front door, the first rear door, and the second rear door. OFFS 1, OFFS 2, OFF3 The number of alighting passengers the bus stop at the stop used front door, the first rear door, and the second rear door. LOAD AT DEPARTURE The number o f passengers on the bus as the bus leaves or passes the stop STOP DISTANCE Travel distance between two consecutive stops STATTTIME Stationary time: The time the bus not in motion; vary from 10 to 45 seconds TRIP ID Unique index associated with a trip STOP ID Unique index associated with a bus stop TRIP STATUS Start or end a trip MATCH Unique index associated with a pattern in each pick data LA 7 GPS Latitude* ( As of August 2005) LON GPS Longitude* (As o f August 2005) Note: * Not available Another type of data that records the passenger activity at bus doors is also collected by the OC Transpo. Figure 3.6 presents an example of such data. There are 6780 and 2800 recorded observations for bus route 95 and bus route 1 for all selected stops, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 mm w jd m m m ukm e iwch K m c BfSK sn n w iA LW iR ns ieu 9 om tsm m mi SB m m S X £ m MU it 35 IB It 13 8 0 4 4 S Figure 3.6: Structure of APC Data on Passenger Activity 3.4.4 Data Characteristics Before developing the model, it is necessary to analyze the acquired data, so that we can obtain the general characteristics of arrival times and dwell times at each bus stop in different period of time. In order to do that, descriptive statistics were applied. 3.4.4.1 Arrival time Arrival time was analyzed in term of bus on-time performance, measuring the differences between the schedule and actual arrival. The data were collected at each bus stop and clustered by time of day and seasonal time (e.g., SEASON 1, SEASON 2, SEASON 3, and SEASON 4). A positive value of difference indicates the bus was behind the schedule, or lateness, while a negative one represents earliness. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two periods of time in the year 2005 were chosen to consider the effects of weather on bus on-time performance. The WINTER data set (Jan 3rd to Apr 23rd, 2005) represents winter conditions and the SUMMER data set (Jun 26th to September 4th, 2005) accounts for summer conditions. The means and standard deviations of lateness and earliness of buses on bus route 1 in the two seasons are shown in Tables 3.6 and 3.7 while that of bus route 95 are shown in Tables 3.8 and 3.9. Table 3.6: Means and Standard Deviations of Bus on-time Performance in the winter. Bus Route 1 (Greensboro - Rideau) Standard Number of Time Period Stop Name Mean (minute) Deviation observations (Minute) GREENSBORO 1A -0.02 1.38 51 BANK-JOHNSTON NS -0.88 1.48 49 BANK- WALKLEY -1.59 1.47 56 8 a.m.-lO a.m. BILLINGS BRIDGE 4C 0.06 1.90 67 (WINTER) BANK -HOLMWOOD 2.02 2.40 78 BANK- GLADSTONE 4.71 2.55 81 RIDEAU 3A 3.17 3.43 53 GREENSBORO 1A 1.07 2.15 45 BANK-JOHNSTON NS -0.63 2.66 43 BANK- WALKLEY -0.64 3.02 42 12 noon -2 p.m. BILLINGS BRIDGE 4C 1.05 2.71 42 (WINTER) BANK -HOLMWOOD 1.63 3.11 40 BANK- GLADSTONE 4.65 3.72 40 RIDEAU 3A 4.87 3.84 49 GREENSBORO 1A 0.67 1.39 20 BANK-JOHNSTON NS -0.87 3.03 23 BANK- WALKLEY -0.83 2.93 24 4 p.m. - 6 p.m. BILLINGS BRIDGE 4C 0.54 3.16 24 (WINTER) BANK -HOLMWOOD 2.42 3.73 24 BANK- GLADSTONE 4.52 3.80 23 RIDEAU 3A 4.17 4.61 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 Table 3.7: Means and Standard Deviations of Bus on-time Performance in the Summer. Bus Route 1 (Greensboro - Rideau) Standard Number of Time Period Stop Name Mean (minute) Deviation observations (Minute) GREENSBORO 1A 0.66 1.21 32 BANK-JOHNSTON NS -0.91 1.81 33 BANK- WALKLEY -1.88 1.95 34 8 a.m.-10 a.m. (SUMMER) BILLINGS BRIDGE 4C -0.27 1.99 49 BANK -HOLMWOOD 2.17 2.73 48 BANK- GLADSTONE 5.23 3.39 47 RIDEAU 3A 3.55 3.73 45 GREENSBORO 1A 0.96 2.33 28 BANK-JOHNSTON NS -0.41 3.07 27 BANK- WALKLEY -0.11 3.39 27 12 noon -2p.m. (SUMMER) BILLINGS BRIDGE 4C 1.26 3.38 27 BANK -HOLMWOOD 1.48 2.67 27 BANK- GLADSTONE 5.30 3.39 27 RIDEAU 3A 3.55 4.82 27 GREENSBORO 1A BANK-JOHNSTON NS -0.60 1.82 21 BANK- WALKLEY -0.5 1.93 20 4 p.m.-6p.m. (SUMMER) BILLINGS BRIDGE 4C 0.95 1.68 19 BANK -HOLMWOOD 1.29 2.03 18 BANK- GLADSTONE 3.65 2.17 18 RIDEAU 3A 0.94 2.01 33 Table 3.8: Means and Standard Deviations of Bus on-time Performance in the Winter. Bus Route 95 (Blair - Lebreton) Time period Standard Number of Stop Name Mean (minute) Deviation Observations ( minute) BLAIR 2B 0.37 3.02 119 ST LAURENT 2B 0.55 2.65 126 TRAIN 2A 0.79 2.34 122 8a.m.-10 a.m. LEES 2B 1.10 (WINTER) 2.31 126 MACKEZINE KING 2A 2.10 2.64 128 ALBERT -KENT 2.36 2.78 129 LEBRETON 0.95 3.04 134 BLAIR 2B -0.59 1.66 111 ST LAURENT 2B -0.27 1.70 111 12 noon-2 TRAIN 2A -0.03 2.63 107 p.m. LEES 2B -0.18 1.73 119 (WINTER) MACKEZINE KING 2A 0.68 1.80 115 ALBERT -KENT 0.99 1.98 138 LEBRETON -0.40 2.12 110 BLAIR 2B -0.11 1.72 138 ST LAURENT 2B 0.40 1.81 138 TRAIN 2A 0.94 1.94 134 4 p.m.-6 p.m. LEES 2B 0.97 2.04 143 (WINTER) MACKEZINE KING 2A 2.76 2.88 138 ALBERT -KENT 3.20 2.50 143 LEBRETON 0.59 2.66 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 Table 3.9: Means and Standard Deviations of Bus on-time Performance in the Summer. Bus route 95 (Blair - Lebreton) Time period Stop Name Mean Standard Number of ( Minute) Deviation Observations (Minute) BLAIR 2B 0.17 0.162 98 ST LAURENT 2B 0.47 1.68 95 TRAIN 2A 0.78 1.67 98 8 a.m.-10 a.m. 0.85 1.80 99 (WINTER) LEES 2B MACKEZINE KING 2A 1.71 1.87 103 ALBERT -KENT 2.28 2.16 102 LEBRETON 0.61 2.43 103 BLAIR 2B -0.19 1.64 59 ST LAURENT 2B 0.43 1.70 61 TRAIN 2A 0.97 1.67 61 12 noon -2p.m. (WINTER) LEES 2B 0.62 1.55 63 MACKEZINE KING 2A 1.43 1.65 61 ALBERT-KENT 2.08 2.11 64 LEBRETON 0.63 2.52 60 BLAIR 2B 0.67 1.97 80 ST LAURENT 2B 1.27 1.96 83 TRAIN 2A 1.78 2.09 81 4 p.m-6 p.m. 2.27 79 (WINTER) LEES 2B 1.88 MACKEZINE KING 2A 3.26 2.57 79 ALBERT -KENT 3.63 2.88 80 LEBRETON 0.90 2.87 84 From the tables, we can see that, in the same time period, the lateness of the stops located in the CBD area are always longer than that of the stops located outside the CBD. For route 1, the means of times decrease in the afternoon both in the winter and summer, implying that the adherence to the schedule is better in the afternoon. In contrast, for route 95, the better adherence to the schedule is found in the middle of the day. Further analyses were carried out in the form of two-sample-t test technique. The means of bus on-time performances at the same bus stop were compared in terms of different time periods (e.g., 8 a.m. to 10 a.m., 12 noon to 2 p.m., and 4 p.m. to 6 p.m.) and various seasons (e.g., the winter and the summer). The results of the tests for bus Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 route 95 are shown in the Tables 3.10 to 3.12, and those of bus route 1 are presented in Tables 3.13 to 3.15. Table 3.10: Two-sample t statistics for Winter and Summer Time Route 95- Morning period (8 a.m. to 10 a.m.) Ho: m1=m2 Ha: m jf m2 Two-sample t values ( 2 tail-test) STOP ti dfl T, BLAIR 2B 0.721 98 1.984 ST LAURENT 2B 0.268 95 1.984 TRAIN 2A 0.037 98 1.984 LEES 2B 0.912 99 1.984 MACKEZINE KING 1.312 103 1.984 ALBERT-KENT 0.307 102 1.984 LEBRETON 0.957 103 1.984 Notes for tables (3.10) to (3.12) t] = t values resulted from the comparisons the means measured in the winter time to the mean measured in the summer time , given the same time period of day. dfl = Degree of freedom corresponding with /, TI = Critical T values with 0.05 level of significance corresponding with dfl. Underlined values: Not significant at 0.05 level or cannot reject Ho. (t< T) Table 3.11: Two-sample t statistics for winter and summer time Route 95- Noon period (12 a.m. to 2 p.m.) Ho: mj=m2 Ha: m2 Two-sample t values (2 tail-test) STOP ti d f Ti BLAIR 2B 1.507 59 2.001 ST LAURENT 2B 2.684 61 2.000 TRAIN 2A 3.726 63 2.000 LEES 2B 1.749 61 2.000 MACKEZINE KING 2.779 63 2.000 ALBERT-KENT 3.482 61 2.000 LEBRETON 2.689 64 2.000 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 Table 3.12: Two-sample t statistics for Winter and Summer Time Route 95- Afternoon period (4 p.m. to 6 p.m.) Ho: mi=rri2 Ha: mf£ m2 Two- sample t values ( 2 tail-test) a l U r ti dfi T 1 BLAIR 2B 2.949 80 1.990 ST LAURENT 2B 3.287 83 1.990 TRAIN 2A 2.933 81 1.990 LEES 2B 2.963 79 1.990 MACKEZINE KING 1.319 79 1.990 ALBERT-KENT 1.120 80 1.990 LEBRETON 0.816 84 1.990 For route 95 (i.e. the Transitway bus route), as shown in Tables 3.10 to 3.12, almost all of the means based on collected data at bus stops during the noon period (12 noon to 2 p.m.) and during the afternoon period (4 p.m. to 6 p.m.) are statistically significant in terms of difference between winter and summer at 0.05 level of significance. This means that the buses of route 95 adhered to the schedule differently depending upon seasons during such time periods. However, for the morning period (8 a.m. to 10 a.m.), such differences were not found. In other words, during the morning period, there is no difference in bus on-time performance between the winter and the summer. Also, for bus route 1 (i.e. mixed traffic bus route), the test results in Tables 3.13 to 3.15 show that at 0.05 significance level, the null hypothesis (i.e., the means measured in the winter time are the same as the means recorded in the summer time) cannot be rejected, for any time period of day. These imply that there are no statistical differences in bus adherence to schedule between the winter and the summer time for the mixed-bus route. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 Table 3.13: Two-sample t statistics for Winter and Summer Time ______Route 1- Morning period (from 8 a.m. to 10 a.m.) ______Ho: mi=m2 Ha: m if m2 Two-sample t values ( 2 tail-test) STOP ti dfi T] GREENSBORO 1A 2.359 32 2.038 BANK-JOHNSTON NS 0.079 33 2.036 BANK- WALKLEY 0.748 34 2.034 BILLINGS BRIDGE 1.978 49 2.009 BANK -HOLMWOOD 1.327 48 2.011 BANK- GLADSTONE 1.813 47 2.013 RIDEAU 3A 0.521 45 2.015 Notes for tables (3.13) to (3.15) tj = t value that is resulted from the comparisons the means measured in the winter time to the mean measured in the summer time, given the same time period of day. dfi = Degree of freedom corresponding with tj TI = Critical T values with 0.05 level of significance corresponding with dfl. Underlined values: Not significant at 0.05 level or cannot reject Ho. (t Table 3.14: Two-sample t statistics for winter and summer time ______Route 1- Noon period (12 a.m. to 2 p.m.) ______Ho: mi=m2 Ha: m f m2 Two-sample t values ( 2 tail-test) STOP ti dfl T, GREENSBORO 1A 0.202 28 2.048 BANK-JOHNSTON NS 0.307 27 2.052 BANK- WALKLEY 0.661 27 2.052 BILLINGS BRIDGE 0.272 27 2.052 BANK -HOLMWOOD 0.211 27 2.052 BANK- GLADSTONE 0.740 27 2.052 RIDEAU 3A 1.225 27 2.052 Table 3.15: Two-sample t statistics for winter and summer time ______Route 1- Afternoon period (4 p.m. to 6 p.m.) ______Ho: mi=m2 Ha: m/fm2 Two- sample t values ( 2 tail-test) STOP ti dfi Ti GREENSBORO 1A BANK-JOHNSTON NS 0.356 21 2.080 BANK- WALKLEY 0.444 20 2.086 BILLINGS BRIDGE 0542 19 2.093 BANK -HOLMWOOD 1.257 18 2.101 BANK- GLADSTONE 0.929 18 2.101 RIDEAU 3A 3.440 28 2.048 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 In order to test if the on-time performance of the buses varies depending upon the different time periods of the same day for a given season, the means of on-time performance collected at bus stop during each time period were also compared by the same technique described above. The test results are shown in Tables 3.16 and 3.17 for bus route 95, and Tables 3.18 and 3.19 for bus route 1. Table 3.16: Two-sample t statistics in Different Time Periods of Day Route 95- The winter time (the WINTER data set) Ho: mi=m2 Ha: m /f m2\ Homi=m3 H a’: m m 3 STOP Two-sample t values ( 2 tai1-test) h df2 t 2 t3 dfi t 3 BLAIR 2B 3.014 110 1.984 1.533 119 1.984 ST LAURENT 2B 2.867 110 1.984 0.532 126 1.984 TRAIN 2A 2.477 107 1.984 0.555 122 1.984 LEES 2B 4.926 119 1.984 0.486 126 1.984 MACKEZINE 4.628 115 1.984 1.950 128 1.984 ALBERT -KENT 4.941 138 1.984 2.609 129 1.984 LEBRETON 4.074 110 1.984 1.046 134 1.984 Notes for tables (3.16) & (3.17): Underlined values : Not significant at 0.05 level or cannot reject Ho or H f. (t< T) h, 13 = Values that are resulted from the comparisons between the means measured the morning period (8 A.M. to 10 A.M.), the means measured in noon period (12 A. M. to 2 P.M.), and those of afternoon period (4 P.M. to 6 P.M.), respectively. df2, dfi = Degree of freedom corresponding with t2t t} T2, T3 = Critical T values with 95 percent degree of confidence corresponding with df2, df3. For bus route 95, the test results show clear differences between the means of on-time performance measured in different time periods of day. As shown in Table 3.16, during the winter time, there are statistically significant differences of bus on-time performance between morning and noon time period. For the summer time, the statistical differences of the means are found between morning and afternoon period (Table 3.17). For bus route 1, the differences between the means are not as clear as those of bus route Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 Table 3.19: Two-sample t statistics in Different Time Periods of Day Route 1- The summer time (the SUMMER data set) Ho\ ml=m2,Ha: m2; H o’: mj=m3Ha: m3 STOP Two-sample t values ( 2 tai -test) h df 2 t 2 h df 3 Ts GREENSBORO 1A 0.613 28 2.048 BANK-JOHNSTON NS 0.747 27 2.052 0.611 21 2.080 BANK- WALKLEY 2.414 27 2.052 2.528 20 2.086 BILLINGS BRIDGE 2.155 27 2.052 2.547 19 2.093 BANK -HOLMWOOD 1.066 27 2.052 1.419 18 2.101 BANK- GLADSTONE 0.086 27 2.052 2.209 18 2.101 RIDEAU 3A 0.00 27 2.052 2.633 28 2.048 In this subsection, the data on arrival time of the bus were investigated thoroughly. Based on the results of two-sample-t tests, some conclusions on the bus arrival times of the two bus routes can be drawn. - For mixed traffic bus route (i.e. bus route 1), the statistical differences of bus on time performance recorded at stops during winter and summer time are not found for all three time periods of the day (i.e. morning, noon, and afternoon). This implies that the winter does not have much influence on punctuality of the buses on a mixed bus route. They were always behind schedule. However, for Transitway (i.e. bus route 95), such differences were found for middle of the day and the afternoon. - For the same season and for the same day, bus on-time performance was different, depending upon time periods and type of bus routes. For bus route 95, these differences were very clear. At almost all the selected stops of this bus route, the differences can be found between morning time and noon time during the winter time and between morning and afternoon time in the summer time. In the winter time, buses tend to be on-time in the noon time. During summer time, buses tend to arrive to stop later in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 the afternoon. For bus route 1, the differences were not clear. On the average, buses were behind schedule by the same amount. 3.4.4. 2 Dwell Time and Passenger Activities According to the OC Transpo, dwell time is the time counted from door opening time to the moment when the last passenger activity is finished. In order to eliminate the abnormal dwell time that could have bad effects on the results of the analysis, the study deleted dwell times of over 180 seconds (3 minutes). According to OC Transpo, these are regarded as “unusual events” and therefore these data should not be included in the analysis. Figures 3.7 to 3.12 present the histograms of dwell times based on one year’s observations of each time period of day for each bus route. 4 0 0 Figure 3.7: Dwell time Histogram for Route 1-Time period (8 a.m. to 10 a.m.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 Figure 3.8: Dwell time Histogram for Route 1- Time period (12 a.m. to 2 p.m.) Figure 3.9: Dwell time Histogram for Route 1- Time period (4 p.m. to 6 p.m.) s Figure 3.10: Dwell time Histogram for Route 95- Time period (8 a.m. to 10 a.m.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 M e a n — 1 4 . 1 7 8 Std. Dev. — 15.8824 N — 1.843 60.00 90.00 DWI :LL TIME Figure 3.11: Dwell time Histogram for Route 95- Time period (12 a.m. to 2 p.m.) As shown in the histograms, route 95 has smaller means and standard deviations than that of route 1. As described previously, these differences reflect type of bus used and route characteristics. This means that its dwell time per stop are smaller and more stable than that of bus route 1. The average dwell time per stop tends to be longer in the noon and afternoon compared to that of morning in both types of bus routes. It varies in the range of 12 to 17 seconds for route 95 and 19 to 21 seconds for route 1. 5 0 0 - a>cr l £ 2 0 0 —I 1 0 0 - Mean = 17.0535 1--- Std. Dev. = 16.04315 100.00 150.00 N = 2,839 D W ELL TIME Figure 3.12: Dwell time Histogram for Route 95-Time period (4p.m. to 6 p.m.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 Passenger activity is composed of the passenger movements through the bus doors. Most of the buses in route 95 are low-floor articulated buses with 3 doors while the buses running on route 1 are mostly rigid-body buses. Boarding passengers are required to board at front door while alighting passengers are allowed to get off the bus at all doors. However, boarding passengers can also get on an articulated bus at the rear doors in the peak-hours. Tables 3.20 to 3.25 present the descriptive statistics of passenger activities on each bus route. As shown in the tables, the number of passengers using front door for boarding is about 2.5 times larger than that of using rear doors on average. By comparing the variables OFF 1 ( the number of alighting passengers using front door) in route 1 and route 95, we can see that the ratios between the number of passengers using rear doors and the front door for alighting are about 1.2 to 1.5 for route 1, but about 5.8 to 6.3 for route 95. Obviously, passengers in rigid body buses tend to use front door for alighting more frequently compared to that of articulated buses. Table 3.20: Descriptive Statistics of Passenger Activities. Route 1- Time period: (8 a.m. to 10 a.m.) Variable N Minimum Maximum Mean Std. Deviation ON_1 1056 .00 54.00 4.3163 6.62927 OFF_1 1056 .00 19.00 .9489 1.89468 OFF_2 1056 .00 19.00 .1913 1.12758 OFF_3 1056 .00 37.00 1.5426 3.72237 TOTALON 1056 .00 54.00 4.3920 6.72462 TOTAL_OFF 1056 .00 55.00 3.0123 5.66329 LOADARR 1056 .00 71.00 19.2983 16.29168 TOTAL_PAS 1056 .00 57.00 7.5616 9.02708 LOAD_DEP 1056 .00 65.00 19.5019 16.34428 Note: Variables have been defined in Table 3.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 Table 3.21: Descriptive Statistics of Passenger Activities. Route 1- Time period: (12 a.m. to 2 p.m.) Std. Variable N Minimum Maximum Mean Deviation ON_1 1072 .00 29.00 4.0606 5.31589 OFF_1 1072 .00 19.00 1.3302 2.48977 OFF_2 1072 .00 16.00 .2407 1.41945 OFF_3 1072 .00 27.00 1.7043 3.69738 TOTAL_ON 1072 .00 29.00 4.0886 5.31760 TOTAL_OFF 1072 .00 37.00 3.2752 5.99532 LOAD_ARR 1072 .00 56.00 15.3330 13.56143 LOAD_DEP 1072 .00 56.00 16.1465 13.76907 TOTAL_PAS 1072 .00 44.00 7.3638 8.80063 Table 3.22: Descriptive Statistics of Passenger Activities. Route 1- Time period: (4 p.m. to 6 p.m.) Std. Variable Observations Minimum Maximum Mean Deviation ON_1 648 .00 33.00 4.5077 6.21833 OFF_1 648 .00 14.00 1.4105 2.46672 OFF_2 648 .00 17.00 .1867 1.35414 OFF_3 648 .00 19.00 1.7160 3.27176 TOTAL_ON 648 .00 33.00 4.5556 6.22082 TOTALOFF 648 .00 29.00 3.3133 5.48954 LOAD_ARR 648 .00 59.00 14.7469 13.32245 TOTALPAS 648 .00 50.00 7.8688 9.29774 LOAD_DEP 648 .00 58.00 15.9892 13.47623 Table 3.23: Descriptive Statistics of Passenger Activities. Route 95- Time period (8 a.m. to 10 a.m.) Variables Observations Minimum Maximum Mean Std. Deviation ON_1 2039 .00 21.00 2.7533 2.94677 ON_2 2039 .00 15.00 .8524 1.74539 ON_3 2039 .00 15.00 .8818 1.76050 OFF_1 2039 .00 21.00 .7396 1.59007 OFF_2 2039 .00 27.00 2.1918 2.98886 OFF_3 2039 .00 40.00 2.4998 3.38369 TOTALON 2039 .00 37.00 4.4875 5.33277 TOTAL_OFF 2039 .00 79.00 5.4311 7.14610 LOAD_DEP 2039 .00 97.00 30.2325 20.35251 LOADARR 2039 .00 96.00 31.1761 17.77509 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 Table 3.24: Descriptive Statistics of Passenger Activities. Route 95- Time period (12 a.m. to 2 p.m.) Variables N Minimum Maximum Mean Std. Deviation ON_1 1843 .00 23.00 3.6316 3.76717 ON_2 1843 .00 17.00 1.3820 2.37809 0N_3 1843 .00 15.00 1.3006 2.32106 0FF_1 1843 .00 15.00 .9078 1.58890 OFF_2 1843 .00 23.00 2.6088 3.33336 OFF_3 1843 .00 22.00 3.0222 3.96141 TOTALON 1843 .00 41.00 6.3142 7.11558 TOTAL_OFF 1843 .00 48.00 6.5388 7.96463 LOAD_ARR 1843 .00 88.00 29.2702 14.47510 LOAD_DEP 1843 .00 98.00 29.0456 16.98444 Table 3.25: Descriptive Statistics of Passenger Activities. Route 95- Time period (4 p.m. to 6 p.m.) Variables N Minimum Maximum Mean Std. Deviation ON_1 2839 .00 33.00 4.0775 4.18456 ON_2 2839 .00 19.00 1.7263 2.66858 ON_3 2839 .00 26.00 1.8859 2.93571 OFF_1 2839 .00 17.00 1.0398 1.93940 OFF_2 2839 .00 26.00 2.9362 4.11590 OFF_3 2839 .00 36.00 3.6404 5.09293 TOTALON 2839 .00 52.00 7.6897 8.05438 TOTAL_OFF 2839 .00 68.00 7.6164 10.34543 LOAD_ARR 2839 .00 104.00 39.6840 19.96143 LOADDEP 2839 .00 106.00 39.7573 21.18925 The average load factors ( the number of passengers in a transit vehicle prior to stopping divided by vehicle capacity) of both bus routes are quite low at about 35% and 52% for route 1 and route 95, respectively. On the average, there are 4 boarding passengers and 3 alighting passengers for each stop of route 1. The corresponding numbers for route 95 are 6 and 7 for each stop. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 3.5 Summary In this chapter, based on the observations of a bus travel time and its components, the framework of the proposed model for bus arrival prediction and its modules were presented. Because moving time accounts for about 60 percent of total travel time and it is vulnerable to delays caused by many unforeseen stochastic traffic-related factors, it is difficult to examine the effect of each factor on moving time separately. Therefore, the proposed model aggregated moving time and relevant delays into running time. By doing so, running time can be obtained from the departure time and the arrival time recorded by the APC and AVL systems between two consecutive bus stops. Dwell time is predicted separately from running time so that the model can capture the effect of long running times to long dwell time at bus stops, which may cause the bus to be behind the schedule at the stops thereafter. The mechanism of the whole model with its three constituent modules was presented in order to explain how the model predicts real-time bus arrivals. Before going to develop the prediction model, the APC and AVL data were collected from the OC Transpo and processed. To understand the preliminary operational characteristics of the two bus routes, namely route 95 and 1, which were selected for modelling purposes, several descriptive statistics were discussed. In the next chapter, the first module of the model will be described. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 BUS RUNNING TIME PREDICTION MODULE 4.1 Introduction Observations on bus travel time components presented in the previous chapter show that running time is the key component in overall terms. Hence, if a model makes poor predictions of bus running time, this will cause the failure of the overall model itself. For this reason, almost all bus arrival prediction models and algorithms reported in literature paid most attention to running time prediction. Also, almost all of these models tried to incorporate most influencing factors in their predictions. In theory, on the basis of all influencing factors, it should be possible to make better predictions. In practice, however, this has been a difficult task because of the data availability and technical constrains. Data availability is of most concern when a researcher attempts to develop a prediction model, given that the quality and quantity of data influences model development in a major way. Data can be divided into static data and dynamic data. The static data such as number of stops, number of intersections, link length, curve radius and so on, of course, are the simplest data that can be collected by common measurements. However dynamic data, whose characteristics vary over time such as densities, volumes and speeds of traffic stream, proportion of left turn and heavy vehicles, are not always 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 available and collectable. Although, from a technical perspective, sensors embedded on the bus routes or the traffic cameras can capture these dynamic data; practically, not all the bus routes have sensors and cameras or even if this was the case, not all these data can be transmitted dynamically. This eliminates the applications of the models based on the above data to make real-time predictions where up-to-minute input data are the vital requirements. To eliminate the difficulties of data availability, an innovative algorithm for bus running time prediction is advanced in this research in order to develop a real-time bus arrival prediction model that is based on the available APC and AVL data. The algorithm is based on a statistical pattern recognition technique where the bus operational state in the past that is most similar to a current bus state reported by the APC-AVL systems is recognized and used to predict the running time of the bus. The methodology to apply the technique to bus running time predictions and the detailed procedure are presented in the following sections. 4.2 Problem Statement Running time is a stochastic process influenced by many stochastic factors. Hence, capturing the distribution of running times with these factors by using specific probabilistic formulations is difficult. No publication reporting an examination of such distributions has been found in the published literature. The duration of time that a bus has just spent from one stop to another, by itself, reflects the sum of influences including traffic conditions. For example, if the bus was unusually behind the schedule, it could be due to an incident while the bus was running Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 on the link. If the bus was late in the usual manner, this could be due to traffic surges during rush hours. If the bus was earlier than schedule time, it could be due to travel in light traffic. Intuitively, the buses running on the same link and in short headways under similar traffic conditions will tend to have similar running situations on that link. In other words, running times of buses observed around a specific time in the past, say t, may contain information about the running time a bus will spend at time t in the future. This standpoint stimulates the applications of pattern recognition to bus running time predictions. A pattern can be seen as a set of measurements or observations represented in a vector or matrix notation (Schalkoff, 1992, p.6). According to this definition, running times of the buses, together with other traffic data, around a specific departure time of a bus under consideration, can be considered as a pattern for the feature of interest (i.e. running time) of this bus at the time in question. In this research, the statistical pattern recognition technique was chosen to develop a model for predicting bus running times. Specifically, a nonparametric regression technique is proposed for the recognition of the bus running situations in the past that exhibit the closest similarity to the current bus running situation. These, the local neighbours, then will be used to estimate the running times of the buses. 4.3 Nonparametric Regression Method in Statistical Pattern Recognition Nonparametric regression (NP) methods, or kernel-based methods, are the new developments in statistical pattern (SP) recognition (Schalkoff, 1992). They stem from curve smoothing techniques and non-linear time series methods defined in the 1970s. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 The term nonparametric was used to differentiate these methods from the parameter regression ones where the data were used to build a series of equations with parameters resulting from parameter fittings. Various prior relations among data variables have been defined such as linear, logistic, log-linear, and so on, depending upon the data. The nonparametric regression methods, in contrast, do not need any specific prior relation among data. Hence, there are no parameter estimates in NP regression. Instead, to forecast, the NP regression retains the data and searches through them for past similar cases. This strength makes NP regression to be a powerful method due to its flexible adaptations in a wide variety of situations. Another advantage of the NP regression over the traditional parametric regression is that because of no parameter estimates, NP regression can adapt easily with new data added to the model while parameter regression method needs expensive parameter re-computations. This is an especially important characteristic of NP for real time applications to predictions where new data are added to the model continuously. Like artificial neural network (ANN) models, the NP regression models can be trained from actual data; but unlike an ANN model which is considered as a “black-box” method, the NP regression provides statistical inference and confidence interval. There are three popular types of nonparametric regression (Andersen, 2004). - Locally Weighted Scatter plot Smoothing (LOWESS): Model fits data by local polynomial regressions and join them together. This type is most widely used in many application fields. Smoothing Splines: Models partition the data and fit separate piecewise regression to each section, smoothing them together where they join. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 Generalized additive models: The extensions of the types above to specific situations such multiple and multivariate regression, and to non-quantitative dependent variables. In this study, the use of LOWESS method is proposed as the basis for model development. The choice of this method over other methods such as spline regression is based on the task at hand. Our interest is with pointwise estimates and not with the continuous regression curve. The details of LOWESS method are discussed below. 4.3.1 LOWESS Estimation Method 4.3.1.1 The General Functions LOWESS method was first introduced by Cleveland (1979) and further developed for multivariate models (Cleveland et al., 1988). In the 1979 publication, Cleveland proposed the model for bivariate data (Xj, Yi),..., (Xn, YJ which are the independent and identically distributed (i.d.d) samples drawn from population (X, Y). Given a value defined by xa, the estimated value of Y]X= xa is calculated by the regression function m (x). Cleveland supposed that function m(x) can be expanded by Taylor series expansion in a neighbourhood of x0 as follows: m m{x) » m(x0) + m’(x0)(x-x0) + — (x-x0)2+...+ — - (x-xQ)p (4.i) z. p . This polynomial is fitted locally by a weighted least squares regression minimization (Fan et al., 1996). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 if r - A - L fitf, -*>) )%(X, -*) (4.2) t=l 7=« Kh()=K(/h)/h (4.3) Where: h = bandwidth controlling the size of the local neighbourhood Kf, (.) is a Kernel function assigning weights to each datum point. A The solution of Equation 4.2 gives a series of ftJtj=0,..., p . Obviously, from Taylor expansion, we have mj(xo) P j = (4.4) where: m },(x0) is an estimator of m (j>(xq) As ntj (jc0 ) is defined from Equation 4.4, m ®(xo) is also defined to estimate the entire function m ®(). In the matrix notation, Equation 4.2 can be denoted as Min (y -X ft)1 W (y-X ft) (4.5) where: ( K\ 1 X\ x0 . . . ( x l - x 0y ' (Po) Po 1^1 A 1 X 2 - x 0 . . . ( x 2- x 0y y2 A Pi • "0 II x= ■ ; y = ; = . . . A 1 I ■ • • Jn ) yPpj yPp; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 W is the n xn diagonal matrix of weights W = diag{Kk( X , - x 0)} (4.6) The solution of Equation 4.5 can be written as: P= (XTW XfXrWy (4.7) where: XT is the transpose matrix of matrix X A Once is defined, the estimation of Y given X=x0 is easily defined, or a prediction is made. From the general equations, we can see that there is a series of issues that need to be discussed. They are: the bandwidth parameter (or smoothing parameter), the type of Kernel function, the degree of polynomial of the regression function. 43.1.2 Bandwidth Selection Bandwidth plays an important role in the accuracy of the estimation made by LOEWSS. Bandwidth selection is the process of recognizing suitable neighbours around x0, or the focal point (Andersen, 2004). In ST Recognition, the recognition of a point belonging to a neighbourhood of a focal point is usually based on the Euclidian distance from the focal point to that point, and a selected bandwidth. In essence, to find an optimal bandwidth is to balance the trade-off between variance and bias (Fan et al., 1996). A too large bandwidth can create a large modelling bias. In contrast, a very small bandwidth can cause a noisy estimate or large variance. Figure 4.1 depicts the effects of bandwidth on NP regression. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 Several methods have been developed to find an optimal bandwidth given a dataset. Generally speaking, there are two kinds of bandwidth selection: constant bandwidth and variable bandwidth. Constant bandwidth regulates a constant range of x given a focal point while variable bandwidth changes upon the location of the focal point. Constant bandwidth is simple to apply but has a drawback of no neighbour recognition. X Figure 4.1: Bandwidth Selection and NP Regression (,Source: Modified from Fan et al., 1996, p.8) Note: (I) Too small bandwidth (h=0) causes noisy estimation and large modelling variance, (2) Too large bandwidth (h=ao) causes modelling bias, (3 )As h runs from 0 to oo, the resulting estimate runs from the most complicated curve to the simplest curve. ^ Data points • • •o » • B A q Focal point • • O • • • Dependent Variable Dependent Independent variable 4 8 12 16 Figure 4.2: Constant bandwidth Problem Note: I f constant bandwidth is 4 units, no neighbour is found aroundfocal A whereas focal B has 2 neighbours Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 •• • o . £ Data points • Q Focal point • • Dependent Dependent Variable Independent variable 4 8 12 16 Figure 4.3: N-Nearest type of bandwidth Note: IfN nearest neighbours are 3, focal A always has 3 neighbours and so does the focal B Therefore, the N-nearest neighbour type of bandwidth, where N neighbours nearest to a given focal are always recognized, is more common. The difference between the two methods is shown in Figures 4.2 & 4.3. In a study published in 1979, Cleveland proposed to use the distance from the r th nearest neighbour of xq as the defined bandwidth. However, the author did not discuss the value of r that should be chosen. Instead, Cleveland suggested that “ Instead o f thinking in terms o f q, the number ofpoints in the neighbourhood, we think in terms o f f=q/n, the fraction ofpoints in the neighbourhood ”. Because the / values were not suggested, so / was chosen without a certainty that the bandwidth is optimal. Another method to find optimal bandwidth is cross-validation (Hardle, 1990). The main feature of this method is the use of two samples drawn from the same Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 population. The first sample, called the screening sample contains nc data points used for building a regression model. The model then is applied to the predictors for the second sample called the validation sample with nv data points. There are different ways to split nc and nv from population of n to built up different models. Out of those models, there will be several models offering the best average predictive ability. When sample size is small and the time for computing bandwidth is limited, the simplest cross-validation with nv~ l is preferable. This method can also be called the leave-one- out method. In the leave-one-out method presented by Hardle (1990), for each given (xit y j, (n-1) remaining points after removing (xit y t) from the sample are used to build regression A A function m(x) and to predict value of y t , given x, and a bandwidth h. The A values of h then are changed to find the optimal bandwidth h satisfying the equations (Hardle, 1990, pp.148-153) h = argmin[CF(/i)] (4.8) h Where CV (h) is cross-validation function 2 (4.9) Other methods to find the optimal bandwidth are: plug-in-out method (details can be seen in Hardle 1990; Fan et al., 1996; and Loader, 1999), automatic bandwidth and order selection (William 1995; Zhang 2000). These methods are based on the trade-off between variance and bias to find optimal bandwidth automatically. However, these methods are very complicated and difficult to apply in practice. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 4.3.1.3 Weighting Kernel Function Selection The goal of Kernel function is to assign suitable weights in response to each neighbour xL of a focal xq. The kernel function is a continuous and symmetric function which integrates to 1. ^K{u)du = 1 (4.10) There are several possible kernel functions that can be applied for estimation. According to Loader (1999), the selection of Kernel function has small influence on the trade-off between variance and bias of the estimation. Table 4.1: Common Kernel Functions for Univriate data Kernel Function Analytic Form, K(x) Rectangular V2 for |m| <1, 0 otherwise Triangular 1- \u\ for \u\ <1, 0 otherwise 15 2 2 Biweight — (1 — u ) for | u\ <1, 0 otherwise 16 Tri-cube weight (l-\u\s)s for |u < 1,0 otherwise Normal i _ exp(-u2 / 2) f i n Epanechnikov — (1 — u215)145 for \u\<4~5, ) , ) 0 otherwise 4 Note: u = ——— / h= bandwidth. Source: Webb Andrew (2002), p. 109. h 4.3.1.4 Degree of Polynomial Regression The selection of degree of polynomial is the balance between computational ease and the need to reproduce patterns in data. Cleveland (1979) suggested using linear regression for adequate smoothing and computational ease. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 4.4 Modelling Bus Running Time Prediction by Using LOWESS Method and APC - AVL Data Before applying LOWESS method and AVL and APC data for modelling bus running time prediction, it is necessary to determine the parameters for the LOWESS method and the APC-AVL data that will be used in the model. 4.4.1 APC and AVL Data As the purpose of this research is to develop a model capable of predicting running time by using APC and AVL data, it is necessary to discuss the data being used in the model. Data collected by the APC and the AVL systems were shown in Table 3.5. Actual running times from one stop to another stop can be obtained in two ways: subtracting the arrival time of the bus at the stop and the actual departure time of that bus at the stop right before; or summation of the moving time, the stationary and the stop- and-go time of the bus. These running times are used in modelling. 4.4.2 Parameter Selection The role of each parameter in LOWESS method has been presented in the previous parts of this chapter. To apply this method for modelling real-time bus running time predictions, the following parameters and assumptions are used. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 Bandwidth selection method: The leave-one-out method is chosen because of its ease in computation while it can maintain an automatic optimal bandwidth. Hence, it is suitable for real-time estimations. Kernel function selection: The Tri-cube weight function is selected. Degree of polynomial regression function: The linear regression is proposed. 4.4.3 Pattern Selection Assume that bus k leaves stop i at time Dp(k)actuaU We have to predict running time the bus will spend to arrive at the next stop i+1, ro(k). By that time Dp(k)actuau , the running times of the previous buses ( e.g., k-1, k-2,k-d) in the link bounded by stop i and stop (i+1) have been collected. Let us denote these running times as ro(k-l), ro(k- 2),..., ro(k-d). Now, the running time being predicted ro(k) is presented by pattern Ro R 0 = {r0(k-l), r0(k-2),..., r0(k -d )f (4.11. A) Where: d = the number of previous buses that should be taken into account in making the running prediction of the current bus. Also in the past, each running time experienced by bus k in the link (i_i+l) is also presented by its own pattern whose elements are defined similarly above. So we have Ri, R 2,..., Rm patterns, representing the corresponding running times of bus k as recorded in the past n(k), r2(k),...,rm(k). Rj = \rj(k-l), rj(k-2),rj(k-d)]J (4.11.B) (m = number of observations collected in the past,y= 1 to m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 To predict running time of bus k at current time with a given pattern Rq, the following steps are developed to recognize the running times of bus k in the past rj(k ) which have the most similar patterns to pattern Rq. 4.4.4 Pattern Recognition The common way that is applied in SP recognition so as to recognize and classify features in the same group, is based on the Euclidian Distances and a selected bandwidth. Therefore, this step includes two sub-steps: calculate the Euclidian distances from pattern Rq to the other patterns Ri, R 2, —, R m , and recognize the neighbourhood. 4.4.4.1 Euclidian Distance Calculation The Euclidian distance between patterns Ro and Rj (j=l to m) can be calculated as follows: (4.n> 4.4.4.2 Finding Optimal Bandwidth by Leave-One-Out Method A running time recorded in the past rj(k) will be recognized as the neighbour of the one being predicted ro(k), called the focal point, when its Euclidian distance to the focal point is smaller than a regulated bandwidth. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 START M i N=0.1 M i=l O3 £> JS OS *3 Z j=j+l *0a oa Wj\ Equation 4.14 c/3 ViVI oa> uo r i ; Equation 4.15 0 -i i=i+l Errit Equation 4.16 s TT o 3 W) N=N+0.1 M CV(h); Equation 4.17 N<=0.8 M OpNer= argmin CV(h) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 The distances have been defined by Equation 4.12 for all running times. Now we define the bandwidth by using leave-one-out technique presented by Hardle (1990). We have m observations of running times rj(k) from the past and the corresponding pattern for each, Rj(k). Following the algorithm suggested by Hardle, and applying Nadaraya-Watson kernel (shown on the next page) estimation for the sake of shortening the time to find the optimal bandwidth, the following steps shown in Figure 4.4 are pursued. Regulate the initial N-nearest neighbours as 0.1m (for example, if m=100 observations, so N= 10, meaning that 10 neighbours will be around a focal point) and the increment for each step is 0.1m. FOR N= 0.1m, N=N+0.1m TO M=0.8m DO (LOOP1) FOR i=l to M do (LOOP 2) 1. Consider the running time ri(k) as the focal point,( i=l torn). It should be noted that it is represented by a pattern namely Ri (k) = I.n(k-l), n (k-2),..., ri(k-d)]r. 2. Use (m-1) remaining running times to build a N-nearest neighbour regression to predict the value of rt(k), as follows: ■ Calculate the Euclidean distances from the remaining running times to the focal r((k) as: • Calculate the Euclidian distance diSj +[rj{k-2)-r,(k-2)f +...+[rJ(k-d)-rl(k-d)f w ith (i^ j) (4.13) • Arrange the disj in the increasing order Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 • Select the Nth smallest of the arrangement above. The bandwidth is the distance from the focal to the nearest neighbour. Let us call that value as h(N). Calculate the weight by applying tri-cube weight Kernel function: 70 dis . , , W{J) = - 0 - 1---- 1 ) I ro l] (4.14) J 81 'h(N)' l,J Where: I [o, ij = 1 if 0 < | dis j / /i(jY)|< 1 otherwise I [o, i] = 0 For the sake of computational ease, the predicted value of ri(k) made at h(N) is computed by the Nadaraya-Watson kernel estimation. m—- 1i A '•<(*)= (4.15) ■ Calculate the prediction error made at h^) as : Err, (k) = [r, (k) - rt(k) f (4.16) END LOOP 2 3. Calculate the Cross-validation function m CV(h) = ------(4.17) m END LOOP 1 Determine the optimal number of neighbours OpNei = arg MIN{CV(h)} (4-18 A) h Determine the optimal bandwidth Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 H = disQpNei (4.18 B) Where: diSoPNei is the Euclidian distance from the focal point to the smallest Opneih neighbour. 4.4.4.3 Recognition of the Neighbours The Euclidian distances (Eq. 4.12) and the optimal band width H (Eq. 4.18B) have been defined. Hence, given focal point r0 (k) and its pattern R 0 ( see Eq.4.11 A), a point rj (k) will be recognized as the neighbour of rg (k) by applying Tri-cube weight kernel function as follows: Calculate the weight K} = — (1— | — |3)3 I[o, ij (4.19) 81 H Where: I [o, /y = 1 if 0 < |w} ///|< 1 otherwise I [o, q = 0; (j=l to m) From Equation 4.19, we can see that if rj(k) has a distance larger than the bandwidth H, the weight assigned for that point is zero, meaning that rj(k) does not belong to the neighbourhood of focal r0(k). Otherwise, it is a neighbour of r0(k). The influence of this neighbour on the focal point is taken into account by a kernel weight Kj depending upon its location. 4.4.5 Prediction When well defined, the weighted neighbours will be used as independent variables of a polynomial regression in order to estimate the dependent variable which is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 being predicted. In our model, the running time of bus k being predicted is the dependent variable (i.e., ro (k)). The weighted neighbours of this dependent variable are the historical running times recorded in the past (i.e., rj (k)) and recognized by the steps mentioned above. Although a higher degree of polynomial regression results in better predictions, the polynomial regression in this study is proposed to be linear regression for the sake of balancing computational ease and adequate prediction (Cleveland et al., 1979). Equations from 4.2 to 4.7 present the procedure for predicting a univariate variable by using weighted least-square regression minimization. In our case, the prediction is univariate response but its pattern is a multi-dimensional vector. Therefore, the procedure above has been further developed by applying the works of Fan et al., (1996) for multivariate regression as follows. Weighted least square regression function has similar form as for Equation 4.5 min (y-XP)TW(y-XP) (4.20) 1 1 1 h—k h—k 2 W' ri(k - 2) — r0(k - ). . r^(k - d) - r0(k - d) o'* -M 1 1 1 r2(k-2)-r0(k-2). . r2(k - d ) - r0(k - d) o''* Where: X= . .. 1 1 • 1 —k h—k . h—k 1 l—t "- W' rm(k-2)-r0(k-2) ■ rm(k-d)-r0(k-d) £ 1 ( \ \ 'A) Po r2(k) A Px fix * A y = ; P = and the estimated values of beta ft = A \Pd, JPd, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 W is the matrix of weights defined as below ~K t 0 0 0 0 0 ' 0 k 2 0 . . f ff _ 0 0 . . 0 . 0 . . 0 . 0 _ 0 . . . 0 K771 m _ Where: K s ' values are computed in Equation (4.19). - The solution of Equation 4.20 is P = (XTWX)'XT Wy (4.21) The predicted value of ro(k) or the running time that bus k will likely spend in the L IN K ij+ i is ro(k) = fio+ Y J0 J-ro(k - j ) (4*22) y=i 4.4.6 Update Prediction The mechanism of how the bus arrival time prediction model updates its predictions was discussed in the previous chapter. As described earlier, when the bus arrives at the next stop, new data are collected and transmitted to the control center. These data then will be used by this module to predict running times and for all downstream links. The updating process follows the same steps above as noted from pattern recognition to prediction. Each time the bus arrives at a stop, if there are M links between the current stop (i.e. the stop the bus has just arrived) and the last stop, the RTM will be applied M times for predicting the running time that the bus will likely spend on each link. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 4.5 Summary In this chapter, an innovative algorithm to predict running times of bus transit is developed and its details are noted. The methodology applied in this algorithm is based on the LOWESS method, a method in the family of Statistical Pattern Recognition. The reasons for choosing this method are as follows: ( 1) it is a powerful method because it is not based on any specific assumptions on the relationship of variables. (2) The predictions made by LOWESS method are based on the local historical database (both past and current data base) which was recorded around the time the prediction is being made, so the method can capture the surges in traffic stream during peak hours as well as the changes due to traffic incidents. To predict running time by applying LOWESS-based method, four steps were proposed: (1) pattern selection, (2) pattern recognition, (3) prediction, and (4) updating of prediction. Of the four steps, the second step is the most elaborate. This includes a procedure from defining Euclidian distances, finding optimal bandwidth by using leave- one-out method, and recognizing the neighbours. After recognition, the weighted neighbours are used in predicting running times by applying multivariate linear regression. The prediction performance of this module will be presented in chapter 6 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 BUS DWELL TIME PREDICTION MODULE 5.1 Introduction From the transit passenger point of view, one would argue that the predictions of dwell times, departure times and the number of passengers are not what a passenger wants to know. This standpoint is true; however, these predictions contribute to the accuracy of announced arrivals that passengers are interested in. Although dwell time accounts for a smaller portion of total delay (12-26%) compared to that of running time (48-75%), this cannot be neglected, especially when the buses serve high density population areas (Levinson, 1983). As opposed to the above point of view, bus transit providers have considered that data on passenger activities at each stop, and the number of on-board passengers are extremely important for analyzing ridership and the relationship between passenger loading, running times and on-time performance (Pile et al., 1998). In terms of real-time control, the predicted number of passengers boarding and alighting at a stop can help bus dispatchers to have suitable pro-active solutions once a crowded bus stop is detected in advance or the knowledge that bunching is likely to occur. Dwell time is influenced by many factors belonging to two groups. The first group relates to passenger activities such as a number of passengers boarding and 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 alighting, types of passengers (e.g., age, physical health, sex). The second group relates to bus service activities such as types of fare collection, number of doors, seat-capacity, type of bus (e.g., rigid-body bus, articulated bus, low-floor bus or high-deck bus), and service frequencies. Before the invention of the APC system, dwell time analysis was limited due to time consuming and highly labor-intensive manual counting. The APC systems, at present, are creating great potential for studies on dwell time by providing much data not only in quantity but also in quality. This also enables the applications of real-time prediction method for predicting bus passenger activities and analyzing delays. The proposed real-time dwell time prediction module described in this chapter is one part of the entire proposed model developed for predicting bus arrivals. Specifically, predicted dwell times are the chains to link a pair of predicted running times detailed in chapter 4. Thus, this module enables the entire model to predict arrival times and departure times of buses at all stops. This chapter is organized as follows. First, some drawbacks of previous dwell time prediction models are discussed. Second, four proposed sub-modules are then described in detail and integrated in a complete bus dwell time prediction module. Last, the summary closes the chapter. 5.2 Bus Dwell Time Prediction Module 5.2.1 Discussions on the Previous Works A number of problems uncovered in the previous works on dwell time prediction require further discussions: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 As shown in the literature review, it was recognized that the number of boarding and alighting passengers is the vital key to determine the dwell time. However, no discussion on how to predict these numbers is found except in the works of Farhan (2002) and Shalaby et al., (2004). To predict dwell time, the previous models require the values of the number of passenger boarding and alighting at stops. Unfortunately, these values were not available and need to be predicted as well. Therefore, it is reasonable to say that the number of predicted alighters and boarders at a stop is the root of dwell time prediction if one want to know dwell time in advance. Once the number boarding and alighting passengers has been predicted, all the previously noted models can be used to predict dwell time. Otherwise, they can be used only for off-line estimation purposes when the information on passenger activities of buses is available. The assumption that a long headway or a long service frequency can result in a long dwell time at downstream stops was widely accepted and used by many authors (Koutsopoulos et al., 1985; Lin et al., 1999; Ding et al., 2000; Chien et al., 2000; Fu and Liu, 2003; Shalaby and Farhan, 2004). However, the relationship between dwell time and headway was still underestimated. Instead, this assumption is quantified by a simple equation shown below: dwell = sf. apar. tj, (5.1) Where: dwell = Dwell time (sec) s f = Service frequency or headway (sec) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 apar = Average passenger arrival rate (passenger/sec) where arrival rate distribution is usually assumed as a Poisson distribution or uniform distribution; tb = Average boarding time per passenger ( sec/pas). This assumption has some drawbacks: ■ First, as shown in Equation 5.1, these models assume that arrival passenger flow is uniform. This assumption may be erroneous in case of long headways where passengers tend to arrive at stops around the announced arrival times (Abkowitz et al., 1986). ■ Second, based on the assumption above, one could argue that if no passenger comes to the stop during a headway time, the value of average passenger arrival rate will be zero. This will result in a zero value of dwell time, meaning that the bus will not stop at that stop because there is no boarding demand. However, there may be an alighting demand and a stop may be made with a positive dwell time. Hence, the above assumption should be modified to capture this case. It is the fact that boarding passengers at the front door usually wait for the last alighting passenger before getting on the bus. If the bus has two doors and some passengers get off at the front door, dwell time may be lengthened. Therefore, the passenger decision on taking front door for getting off the bus will play an important role in predicting dwell time. If the bus has more than 2 doors (e.g., articulated bus), where passengers can board in rear doors, the dwell time will be the time for serving the busiest door. However, which door Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 will be the busiest door in the above cases and what is the role the passengers play in determining dwell time are still open questions. 5.2.2 The Dwell time Prediction Module (DTM) The DTM proposed in this study is dynamic in a manner that it is based on the real-time predicted boarding and alighting passengers at a stop. In order to do this, two of its sub-modules, namely Real-time Boarding Passenger Prediction (RPSM) and Real time Alighting Passenger Prediction (RASM), have been developed separately. These predictions then are used as the inputs for two other sub-modules called Regression (RESM) and Busiest door (BDSM). The RESM is based on the collected data from AVL and APC systems in the OC Transpo. In this sub-module, the relationship between dwell time and a series of variables (e.g. the number of passengers, bus type, on time performance, and so on) is examined in the form of several statistical regressions. The BDSM is based on the equation suggested by HCM 2000 (Eq. 2.2) with modifications in order to find the busiest door. By combining the sub-modules, two methods (A&B) are proposed in this research for predicting dwell time. Method A is the combination of BPSM, APSM, and RESM. Method B includes BPSM, APSM, and BDSM. Figures 5.1 and 5.2 depict the frameworks of the two methods. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 Boarding Passenger Prediction Sub-Module (RBSM) Predicted if of total boarding passengers APC and AVL Regression Predicted Data Sub-Module Dwell Time (RESM) Predicted ft of total alighting passengers Alighting Passenger Prediction Sub-Module (RASM) Figure 5.1: Dwell time Prediction Module. Method A Boarding Passenger Prediction Sub-Module (RBSM) Predicted # of total boarding passengers Predicted # of APC and AVL Busiest Door boarding and Data Sub-Module alighting (BDSM) passengers at the busiest door Predicted # of total alighting passengers Predicted Dwell Time Alighting Passenger Prediction Sub-Module (RASM) Figure 5.2: Dwell time Prediction Module. Method B The details of each sub-module are presented separately in the following sections and the better prediction method is identified. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 5.23 Real-time Boarding Passenger Prediction Sub-Module The LOWES S-based method developed in chapter 4 is used here to predict boarding passengers at a stop. The steps to obtain a prediction on the number of passengers boarding bus (k) at stop (i+ 1 ) are presented below. 5.2.3.1 Parameter Selection The method applied to select parameters in chapter 4 is used in this chapter as follows: leave-one-out method for optimal bandwidth selection; Tri-cube weight function to recognize the neighbours, and Nadaraya-Watson kernel function for prediction. 5.23.2 Pattern Selection As discussed earlier, there is an erroneous assumption on the distribution of passenger arrival flow at a stop. In this research, the author intended to capture the relationship between headway and the number of boarding passengers at a downstream stop without any prior mathematical assumption about this relationship. Assume that bus k leaves stop i at time Dp(k)actuaij. We have to predict the number of passengers boarding on bus k when it arrives at the next stop i+1. Let us denote this variable as bo(k). At time step Dp(k)actuairi, data on the number of boarding passengers on previous buses at stop (i+1) have been collected. We denote these numbers as bo (k-l), b0(k-2),..., b0(k-d). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill Also at that time step, running times of the previous buses (e.g., r0 (k-1), r0 (k-2),..., r0 (k-d)) in the link bounded by stop i and stop (i+1) have been available. We already have the predicted running time that bus k on the LINK This value is r0 (k) in Equation 4.22 (Chapter 4). In order to take the relationship between the bus headway and the number of boarding passengers without any prior assumption on it, we consider the number of boarding passengers on the previous bus at stop i+1 as well as the recorded headway of this bus as the elements of a pattern defining the number of boarding passengers being predicted when bus k arrives at stop i+1. Because the running time of bus k, which has been predicted, may have influence on the number of boarders being predicted, it is also included as an element of the pattern above. Based on the above assumption, we can say that the boarding passengers being predicted bo(k) is presented by a pattern, namely (Bo), whose three elements are the number of boarding passengers recorded from the previous bus, the running time of that bus, and the predicted running time bus k will spend to come to stop i+1. Bo = [b0 (k-1), r0(k-l), r0(k )f (5.2) Also in the past, the number of boarding passengers on bus k at stop (i+1) is also presented by its own pattern whose elements are defined similarly above. So we have B ]f B2,..., Bm patterns, representing the corresponding number of boarders of bus k recorded in the past bt(k), b2(k), ...,bm(k). Bj = [bj (k-1), rj (k-1), rj (k)] T (5-3) (m = number of observations colleted in the past, 7 = 1 to m). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 Because the patterns contain elements measured on different scales, it is necessary to standardize them. We have: SB j - [sbj (k-1), srj (k-1), srj (k)] T (5.4) Where: SBj ( j =1 to m) = standardized pattern of Bj sbj(k-l) = standardized values of b j(k-l);j-l to m sbj (k-\)= bj( } ) ( k - 1) = The mean of the number of boarding passengers on bus (k-1) m 2>#-D /)«-!)= H------(5.6) m Sb(k-1) = Standard deviation of boarding passengers on bus (k-1) m — Sb(k-\) = a ------:------(5-7) m -\ sr0 (k-1) = Standardized value of running time r0 (k-1) rJk - y(k — 1) sr0 ( k - 1) = — ------(5.8) 0 5r(A:-l) srj(k-l)= Standardized values of running times rj (k-1); (j=l to m) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 sr . (k — 1) = —------(5.9) 7 Sr(k-l) Where: - y (k - 1) = Mean of running times of bus k-1 m l o ( * - 1) r (k- 1) = -H------(5.10) m - Sr(k - 1) = Standard deviation of running times of bus k-1 f J[rJ(k-\)-r (k-\)f Sr(k -1) = j =1 (5.11) m — 1 - sr0 (k)= Standardized value of running time r0 (k) - srj (k)= Standardized values of running times rj(k) »•„(*)= (5.12) Sr(k) sr.(k) = rAk) ------~ r ( k) (5.13) ' Sr(k) Where: Y (k) = Mean of running times of bus k Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 114 m l o w f{k) = (5.14) m Sr(k) = Standard deviation of running times of bus k m Sr(k) = } (5.15) m - 1 5.23.3 Pattern Recognition Following standardization, the patterns are used to recognize the neighbours of b 0 (k) represented by pattern B o (Eq.5.2). The automatic procedure to recognize the neighbours around bo (k) is similar to the one presented in Section 4.4.4 of which we replace the r0(k) and rj(k) with their patterns by the focal bo(k) and bj(k) with the corresponding patterns. For the sake of reducing a lengthy expression, this procedure is not shown here. 5.23.4 Prediction To predict the standardized value of boarders s bo (k), we use the Nadaraya- Watson kernel estimation function as follows. (5.16) j=Uj*i Where Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 OpNei is the optimal number of neighbours around b o (k ). Please see equations 4.18A and 4.18 B for the definition of OpNei. Kj is the weight assigned for the j th neighbour of the defined neighbourhood. Please refer to equation 4.19 for the definition of K j A b 0 - The predicted value of b o (k ) 5.23.5 Update Prediction The updating process follows the same steps above as noted from pattern recognition to prediction. If there are M stops between the current stop (i.e. the stop the bus has just arrived) and the last stop, the DTM will be applied M times for predicting the number of boarders on the bus when it arrives at each stop. 5.2.4 Real-Time Alighting Passenger Prediction Sub-module LOWESS-based method is also applied to predict the number of alighting passengers at a stop. The procedure is as shown below. 5.2.4.1 Parameter Selection The following methods are applied to select parameters in modelling alighting passenger predictions: leave-one-out method for optimal bandwidth selection; Tri-cube weight function to recognize the neighbours, and Nadaraya-Watson kernel function for prediction. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 §.2.4.2 Pattern Selection The number of alighting passengers at a stop does not depend on the service frequency. Therefore, the pattern for the number of alighting passengers of bus k that are being predicted at stop i+1, written as a0 (k), is selected as the composition of only the number of alighting passengers of the two previous buses denoted as ao (k-1) and a0 (k-2) A0=[a0(k-l),a0(k-2)]T (5.17) Where: ao (k) = the being predicted number of passengers alighting from bus k at stop i+1. ao (k-1), ao (k-2) = the number of alighting passengers on the two previous buses (k-1) and (k-2) at stop i+1 Also in the past, each number of alighting passengers on bus k at stop (i+1) is presented by its own pattern with its elements defined in a manner similar to the one above. So we have the patterns Ai, A 2 ... A m A j = [a, (k-1), aj (k-2)] T (5.18) Where: j =1 to m; m = the number of observations in the past 5.2.4.3 Pattern Recognition The procedure to recognize the neighbours around ao (k) is similar to the one presented in Section 4.4.4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 5.2.4.4 Prediction Again, we use Nadaraya-Watson kernel estimation function as follows OpNei “•<*>=— <5-19> IX j= h j* i Where OpNei is the optimal number of neighbours around ao (k). Please see equations 4.18 A and 4.18 B for definition of OpNei Kj is the weight assigned for the f h neighbour of the defined neighbourhood. Please refer to equation 4.19 for the definition of Kj A a0 = The predicted value of ao (k) 5.2.4.5 Update Prediction The updating process follows the same steps as noted above from pattern recognition to prediction. Each time the bus arrives at a stop, if there are M stops between the current stop (i.e. the stop the bus has just arrived) and the last stop, the RASM will be applied M times for predicting the number of de-boarders on the bus when it arrives at each stop. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 5.2.5 Regression Sub-Module This sub-module is developed in order to examine the relationships between dwell time and other variables influencing it. It is a component of method A mentioned in Section 5.2.2. Data collected from APC and AVL systems of the (X) Transpo are used to find these relationships. 5.2.5.1 Variable Selection and Preparation The variables selected in the regression models are shown in Table 5.1. Because some variables were not colleted by the APC-AVL systems but can be transformed from other variables, it is necessary to prepare data before going to the regression analyses. The variables were transformed and calculated as shown below. The explanations of variables are in Tables 3.5 and 5.1. LOAD ARR: = (LOAD DEP) + (TOTALOFFS) - (TOTAL ONS) (5.20) TOTAL_ONS- (ON I) + (ON_2) + (ON 3) (5.21) TOTALOFFS = (OFF1) + (OFF2) + (OFF3) (5.22) PUNT = (A CTJ1ME) - (EXPEC TIME) (5.23) TOTALPASS = (TOTALONS) + (TOTALOFFS) (5.24) LF = (LOADARR)/BUSCAPA (5.25) where: BUS_ CAPA is the number of designated seats of transit bus; BUSCAPA = 40 for rigid bus; = 65 for articulated bus. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 Table 5.1: Variable Selections for RESM Variable Type of Code Description of Variable Reason for Selection Variable Boarding The total number of It has a major influence Numeric TOTALONS passengers passengers boarding at a stop on dwell time. Alighting The total number of It has an influence on Numeric TOTALOFFS passengers passengers alighting at a stop dwell time Boarding Total passengers getting on It has a major influence and Numeric TOTALPASS and off at a bus stop on dwell time. alighting This effects the On- board Number of passengers on the circulation in the bus. passengers Numeric LOADARR bus before it arrives or passes Therefore it will prior stop the stop. influence dwell time A value of LF close to 1 Ratio between the number of means that the bus is Loading Numeric LF on-board passengers prior stop likely full. Hence, this factor and the capacity of the bus may cause a long dwell time. Alighting If passengers alight at Number of alighting passengers Numeric OFF 1 front door, it will increase passengers using front door at front door dwell time. A long lateness can cause Measured by lateness and a crowded downstream Punctuality Numeric PUNT earliness of the bus. stop resulting in a long dwell time. Articulated bus or rigid bus Dummy (all are low-floor buses). Dwell time depends on Bus Type BUSTYPE l=Articulated Bus; bus type 0 = Rigid bus Time of day (Morning, noon, Dwell time depends on Time Dummy TIME after noon). 1= Morning; 2= time of day Noon; 3= afternoon Dwell time may be 0= Winter; 1 = Spring; Season Dummy SEASON different for summer and 2=Summer;3= Fall winter Number of Increasing number of Numeric DOORS Self explanatory Doors doors reduces dwell time Stop 1= Stop located at CBD The location of bus stop Dummy STOP LOCA location 0= Otherwise may effect dwell time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 5.2.5.2 Regression Functions Before going to regression analyses, dwell times being used in the regression functions are processed to delete unreliable records. One of these records is the zero values, meaning that the bus does not stop at the bus stops. Such records are not included in the model and being deleted. The other records that should be removed from the dataset are the extreme values of dwell times and outliers. However, deleting these records cannot be done without any consideration. In this study, the records of dwell time over 180 seconds are deleted without any concern because such time durations are usually the layovers. After removing these records, the remaining data set was examined to find additional records that should be deleted. Examinations of the data show that the mean of dwell time recorded for all seasons and for all times of day for bus 95 and 1 are 15.86 and 20.40 seconds, respectively. The records, which are suggested by the SPSS package as the extreme cases and outliers, are the ones with durations of over 44 seconds in bus route 95 and 57 seconds in bus route 1. The boxplots shown in Figures 5.3 and 5.4 present the outliers and extreme cases which are signified by small circles and asterisks1, respectively. According to the SPSS, the cases with values between 1.5 and 3 box lengths from the upper or lower edge of the box are considered as outliers. The cases with values higher than 3 box lengths from both edges of the box are considered as extreme cases. The box length is the interquartile range. The outliers and extreme cases as mentioned above (i.e. cases with recorded dwell time larger than 57 seconds for buses of route 1 and 44 seconds for bus 95) should be 1 The number beside the asterisk (or circle) is the case number Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 erased in order to improve the prediction capability of the regression models. In this study, the dwell times lower than 57 seconds are kept for both routes. The sample size of the database for regression analysis reduces from 9983 cases to 8685 cases after outlier and zero value deletions . 200 . 0 0 - :■ Dwell Time (seconds) Figure 5.3: Boxplot of Dwell time. Bus route 1 7,139 Figure 5.4: Boxplot of Dwell time. Bus route 95 2 It was found that when using all data without any outlier and extreme case deletion for a pilot multiple- regression, the R2 of this equation was rather low. Please see Appendix B l for this regression. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 Type A-l: Simple Linear Regression The general regression equation is shown below. dwell = P0 +P, .(TOTAL _ ON) + P2 .(TOTAL _ OFF) + /?3 .(TOTAL _ PASS) + fa.(LOAD _ARR) + P}.(LF) + +P6.(PUNT) + J37.(BUS _TYPE) + J3g.(TIME)+ ( 5 .2 6 ) P9.(SEASON) + Pl0.(DOORS) + p u.(STOP _LOCA) The backward stepwise regression method is selected in order to track the effect of each variable on dwell time. As a part of this method, variables with significance values of more than 0.15 (i.e., they are statistically not significant) are not included in the model. From this perspective, one may argue that a significance level of 0.05 should be better. Of course, a selection of a low value of level of significance (i.e. high confidence level) will reduce the risk of taking an unimportant variable in the regression model. However, 0.05 is too low and often results in the deletion of the important variables from the model. Therefore, the values of a = 0.15-0.2 are suggested (Scott Menard, 2001, p. 64). The SPSS (Version 13.0) package was used to develop the regression model. Eighty percent of the total cases were drawn randomly. The remaining cases were used for testing the models. The models showing the largest R-square, along with significant t- test for parameters, were selected. The results of the three runs are shown in Tables 5.2 and 5.3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 Table 5.2: Coefficients of the Best Regression Coefficients a Unstandardized Standardized Coefficients Coefficients Model B Std. Error Beta t Sig. 1 (Constant) 15.750 1.154 13.648 .000 TOTALJDNS 1.220 .016 .673 75.097 .000 TOTAL_OFFS .438 .016 .246 27.173 .000 LOAD_ARR .072 .029 .112 2.495 .013 LF -4.935 1.649 -.125 -2.992 .003 TIME .113 .119 .008 .953 .341 PUNT .003 .001 .047 5.102 .000 SEASON .097 .088 .009 1.098 .272 STOP_LOCATE 2.125 .217 .089 9.804 .000 DOORS -4.361 .419 -.159 -10.401 .000 2 (Constant) 16.020 1.119 14.317 .000 TOTALJDNS 1.222 .016 .674 75.854 .000 TOTALOFFS .439 .016 .246 27.183 .000 LOAD_ARR .075 .028 .117 2.617 .009 LF -5.086 1.642 -.129 -3.098 .002 PUNT .003 .001 .047 5.096 .000 SEASON .093 .088 .009 1.055 .291 STOP_LOCATE 2.127 .217 .089 9.816 .000 DOORS -4.385 .419 -.160 -10.478 .000 3 (Constant) 16.175 1.109 14.583 .000 TOTAL_ONS 1.222 .016 .674 75.856 .000 TOTALOFFS .438 .016 .246 27.163 .000 LOAD_ARR .074 .028 .116 2.605 .009 LF -5.064 1.641 -.128 -3.085 .002 PUNT .003 .001 .047 5.116 .000 STOP_LOCATE 2.126 .217 .089 9.812 .000 DOORS -4.397 .418 -.161 -10.511 .000 a- Dependent Variable: DWELL_3 dwell = 16 .175 + 2.126 (STOP LOCATE ) + 0.438 (TOTAL OFFS ) + R 2 = 0.581 ( = 14.583 ( = 9.8 1 2 ~ ( = 2 7 .1 6 3 _ + 1.222 (TOTAL ONS ) - 4.397 ( DOORS ) + 0.074 (LOAD ARR ) (5 '27) ( = 7 5 .8 5 6 — ( = - 10.511 1 = 2 .6 0 5 ~ - 5.064 (LF) + 0.003 (PUNT ) ( = - 3 .0 8 5 ( = 5.116 After 3 runs, the stepwise regression method returned the best regression as shown in Table 5.2. Two out of 9 variables were removed from the model (i.e. TIME and SEASON). All parameters are statistically significant. With R-square of 0.581, the model is satisfactory. Following Equation 5.27, if we control other variables, dwell time will increase by 2.13 seconds if bus stop is located in the CBD area compared to that of a stop outside this Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 area. The number of doors has a strong influence on dwell time. If the bus has 3 doors instead of 2 doors, dwell time can decrease up to 4.34 seconds when other variables are controlled. Dwell time will increase if the bus has more boarding or alighting passengers. Table 5.3: Model Summary for the Best Regression Model Summary A djusted Std. Error of M odel R R S q u a re R S q u a re the Estimate 1 .763a .582 .581 7.66728 2 .763b .582 .581 7.66722 3 .763c .581 .581 7.66729 a- Predictors: (Constant), DOORS, SEASON, STOP_ LOCATE, TIME, TOTAL_OFFS, LF, TOTAL_ONS, PUNT, LOAD_ARR b- Predictors: (Constant), DOORS, SEASON, STOP_ LOCATE, TOTAL_OFFS, LF, TOTALJDNS, PUNT, LOAD_ARR c. Predictors: (Constant), DOORS, STOP_LOCATE, TOTALJDFFS, LF, TOTALJDNS, PUNT, LOAD_ARR Typel A-2: Non- Linear Regression Besides the multiple linear regression developed above, a series of non-linear regression types of function are also examined. - Type A -2.1 : dwell = 0„ + 0 t.(TOTAL ON )+ 0 2j{TOTAL ON)2 + 0, (TOTAL _ OFF )+ 0 4j(TOTAL _PASS) + 0 5 .(LOAD _ ARR ) + 06.(LF)+ 0 7 .(PUNT ) + 0 % .(BUS _ TYPE ) + 0 9 .(TIME ) + V • 1 0 W .(SEASON ) + 0 n .(DOORS ) + 0 n .(STOP _ LOCA ) - Type A-2.2 dwell = 0 O + 0 X.(TOTAL ON) + 02.(TOTAL _ ON)2 + 0 2(TOTAL ON)2 + 0 t (TOTAL _OFF) + 0 S .(TOTAL _ PASS) + 0 b.(LOAD _ ARR) + 0 1.(LF) + 0 t .(PUNT) + 09.(BUS _ TYPE) + (5 *29) 0 lo .(T1ME) + 0 U .(SEASON ) + 0n .(DOORS ) + 0 n .(STOP _ LOCA ) - Type A-2.3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 dwell = /?„ +Pv{TOTALON) + P2.(TOTAL_ONf + P, (TOTAL _OFFS)2 + (5.30) P4(TOTAL_OFF)(TOTAL _ OFF) + P5.(T0TAL _ PASS) + P6.(LOAD_ARR) + Pt (LF) + Pt.(PUNT) + P9.(BUS _TYPE) + pm.(TIME) + Pu.(SEASON) + Pn.(DOORS) + PJSTOP _LOCA) - Type A-2.4 (Gauss Regression) dwell = M l “ *3 exP (-*>2 (TOTAL _ PASS)2)) (5.42) - Type A-2.5 (Weibull Regression) dwell = bx- b 2. exp(-Z >3.(TOTAL_ PASS)b>)) (5.31) - Type A-2 .6 (Morgan-Mercer-Florin Regression) dwell = (bx b 2 + by TOTAL _ PASSbi) /(b2 + TOTAL _PASSb*) (5.32) The types from A-2.1 to A-2.3 are polynomial regressions in term of variables TOTALJDNS and TOTAL OFF whereas types from A-2.4 to A-2.6 are the common non linear regressions where the relationship between dwell time and TOTAL PASS (e.g., total number of passengers getting on and off the bus) is examined. To find the best regression equation for each type, backward regression method was applied for the types from A-2.1 to A-2.3 and the Gauss-Newton method was applied for the rest. By using the SPSS version 13, we can find the best regression equations. Their R-square values and parameters are presented in Table 5.4. Details can be seen in Appendix Bl. Out of the types from A-l to A2-6, type A-2.3 has the highest R-square, meaning that it is the most accurate type as compared with others. As a result, the use of type A- 2.3 is proposed in this research to predict dwell time. Although type A-2.6 is less accurate than type A-2.3, it is a compact one requiring only the total of passenger activities through bus doors recorded or predicted. Therefore, it should be used in case of lack of data on some variables. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 Table 5.4: Proposed Non-linear Regression Models Type The best regression equations R- sq u a re A-2.1 dwell = 13.177+ l.92S(TOTAL ONS) - 0.030 .(TOTAL ONS)2 -0A3%(TOTAL OFF) - 0.830(£F) - (= 21.024 1=51.635 “ ( = - 20.636 (= 28.834 ( = - 2.414 3.922 (DOORS) +1.951 (STOP LOCA) + 0.00\PUNT) 0.610 (=-17342 (=9.684 " (=5.456 dwell = 12.843+ 2.35l(TOTAL ON)- 0.071 (TOTAL ON)1 + 0.00\{TOTAL ONS)1 + A-2.2 ( = 19.804 ( = 33.657 — ( = - 11.873 ” ( = 7.050 - M2%TOTAL OFF) + 0.03 (PUNT) - 0.954 (LF) - 0.153(SEASON) - 4.073 (DOORS) + ( = 28.173 — ( = 5.490 - 2.771 1.821 ( = - 17.953 0.618 UO^STO/5 LOCA) (= 8587 — dwell = 14.434+ 2.0\4(TOTAL ON)- 0.026 .(TOTAL ON)2+Q.5\6(TOTAL OFFS) + ( = 13.921 ( = 54.071 “ ( = - 17.944 “ ( = 15.364 — OJtmiTOTAL OFFS)2 - 0.022 (TOTAL ONSMTOTAL OFFS) + 0.07&.(LOAD ARR) A-2.3 (=3387 — ( = - 11.548 “ — 2.982 - 0.634 - 5.047(LF) + 0.003(PUNT) - 4.748 (DOORS) + 2Ml(STOP LOCA) + OA96(SEASON) -3 3 4 9 ( = 5.492 (= -1 2 3 9 5 (= 10384 — ( = 2.419 A-2.4 dwell = 31.08(1 - 0.795 exp(-0346.(l0~'.TOTAL _ PASS) 2 )) 0.502 A-2.5 dwell = 1184.802 -91.856 exp(-2.557(10 _1 TOTAL _PASS)^°°3) 0.477 dw ell = (-146.259+ 470.907(rOZ4i P ^ ) 0561) /(97.767 + (\Q~'TOTAL PASS)056 A-2.6 0.528 To predict dwell time, the number of boarding passengers (TOTAL ONS) and the number of alighting passengers (TOTAL OFFS), which can be predicted from previous sub-modules (i.e., BPSM and APSM), is used in regression A-2.3. Other related variables (i.e., LOAD ARR, PUNT, and LF) can also be determined indirectly by using Equations 5.20, 5.23 and 5.25. It should be noted that data on all variables that are used in this type are all obtainable from the APC and AVL systems mounted on the buses. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 5.2.6 Busiest Door Prediction Sub-Module If we assume that all passengers board at the front door and alight at the rear- door, the dwell time can be easily predicted as the maximum of the two service times, one for boarding passengers at front door and one for alighting passengers at the rear door. However, data colleted from the APC system by the OC Transpo shows the fact that the on-board passengers alight at all doors while waiting passengers can get on the bus at the front door of a 2 door-bus and from all doors of the 3-door bus. This implies that the dwell time is actually the service time at the busiest door. This can be seen in the suggestions of the Highway Capacity Manual 2000 (Equation 2.2). However, how to find the busiest door was not discussed in literature. In this sub-module, the equation provided by the HCM (2000) is used with changes in finding the most crowded door. It should be recalled that the sub-module is a part of method B, another method proposed in this study for predicting dwell time in real-time, given the predicted number of boarding and alighting passengers. Please refer to Fig.5.2 for details. In this research, dwell time is calculated by using Equation 5.33. dwell = aJa + b.tb + top (5.33) Where: dwell = Predicted dwell time in seconds a = Predicted number of alighting passengers through the busiest door ta = Passenger alighting time per passenger ( follows HCM 2000) b = Predicted number of boarding passengers through the busiest door tb = Passenger boarding time per passenger (follows HCM 2000) top = Door opening and closing time ( 4 seconds) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 In order to find the busiest door for each type of bus, data are clustered upon rigid body bus and articulated bus. The methodologies to find the busiest door for each type are presented below. 5.2.6.1 Rigid-Body Bus A rigid-body bus has two doors, one in front and the other in the rear. At a bus stop, passengers can only get on the bus from the front door while they can get off the bus at any door convenient for them. Before the bus completely opens the doors, alighting passengers tend to move to the doors for ease of exiting. Alighting passengers have priority to get off the bus first. If some passengers get off via the front door, on ground passengers have to wait until the last alighter leaves the bus before getting on. This will lengthen dwell time compared to the case when all alighting passengers use the rear door and all boarding passengers use the front door. For example, assume that there are 8 alighting passengers and 3 boarding passengers. Serving time for each alighter is assumed to be 2.5 seconds and 5 seconds for each boarder. Hence, servicing time to board all 3 passengers is 15 seconds at the front door, and 20 seconds are required for 8 alighting passengers, provided that all alighters use the rear door and all boarders use the front door. The busier door is the rear door (i.e., 20 seconds). However, if 4 out of 8 alighters use the front door, the servicing time at this door will be 25 seconds and for the rear door it is only 10 seconds. The busier door now is the front door instead of the rear door as found in the first case and the extra service time is 5 seconds. Obviously, if Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 passengers get off at the front door, the dwell time is increased. If we can find the busier door, we will have a better prediction of dwell time of the bus. On-board passengers are likely to choose the most convenient door for alighting. Of course, passengers near a door will usually use that door for alighting because of a short distance. Based on the distance and the ease to move to the doors, passengers in the middle of the bus will choose a door for alighting. In general, passengers will use the door that offers them the highest convenience. Intuitively, the convenience of a door can be presented by several variables such as the number of boarding and alighting passengers, the number of standees on the bus, the stop location, time of day, season, and so on. In other words, the likelihood (or the probability) that a door is the busier door depends on an unobserved variable (i.e. the convenience that a door offers passengers at each bus stop) whose values can be quantified by a series of data retrieved from APC and AVL systems. Let us call U as “convenience” function. Assume that we have to relate probability that the front door is the busier door (P front door) with convenience value that the front door offers alighting passengers at bus stops. Obviously, we cannot model this probability directly as Pfront_ door = U, because the values of U may vary outside the range 0 and 1 while the probability must lie between this range. To solve the problem, a commonly used approach in logistic regression is to relate the probability with the odds. Let us assume that the odds fron, d0or describes the ratio of the probability that the front door is the busier door to the probability that it is not so. It is easy to convert back and forth between odds and probability by using Equations 5.34 and 5.35. As shown in the equations, probabilities range from 0 to 1, whereas odds range from 0 to infinity. An Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 odds equal to 1 indicates equal probabilities that the front door is the busier door and it is not so (i.e., 50%). For odds greater than 1, that the front door is the busier door is more likely and vice versa. o d d s f m m J m = J^L_ (5J4) front door _ ° d d s fr0nt door front _ door . » . (5.35) 1 + o d d s p.ont door Where: P frontjoor = the probability the front door is the busier door The natural logarithm of the odds that the front door is the busier door is called the Logit (front_ door). Pfront door Logit (front_ door) = In (oddsfront door) = In (- — = ) (5.36) front _ door Obviously, the variable Logit (front_ door) varies from negative infinity to positive infinity. Therefore, if we model Logit (front_ door) = U, we will have a form of logistic regression. Logit (front door) = U (5.37) We can convert Logit (front_ door) back to the odds by exponentiation. From Equations 5.37 and 5.38 we have O dds jront door 0 (5.38) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 Substitute the o ddsfront_door in Equation 5.38 into Equation 5.35, so the probability that the front door is the busier door can be calculated as below. Pfront door ~ ^ X ~ \ + eu ( 5 . 3 9 ) The probability that the rear door is the busier door ( Prear door) is as shown below Prnr door = P i= -rrir ( 5 . 4 0 ) 1 + e Several types of U equations are proposed in the Equations from 5.41 to 5.43. The regression coefficients of each equation are estimated through an interactive maximum likelihood method. Type B-L.1 U = /?„ +Pt.(TOTAL_ONS) + p 2.(TOTAL _OFFS) + P2.(LOAD _ARR) + Pt .(LF) + fc.(PUNT)+ ( 5 - 4 1 ) P2 .(TIME) + Ps.(SEASON) + P9.(STOP _LOCA) Type B-L.2 U = pn + Pv(TOTAL_ ON) + Pr (TOTAL ON)1 + P,(TOTAL ON)1 + ( 5 . 4 2 ) + pi(i.(LOAD ARR) + P6.(LF) + P7.(PUNT) + P^TIM E) + P9.(SEASON) + Pm.(STOP LOCA) Type B-L.3 U = fia +P .(TOTAL _ ON) + P2 .(TOTAL _ ON)1 + /? , (TOTAL _ OFFS)1 + /?4 (TOTAL _ ONS)(TOTAL _ OFFS) + ( 5 . 4 3 ) P5.(LOAD_ARR) + Pi .(LF) + P7.(PUNT) + Ps.(T1ME) + P,.(SEASON)* Pm.(STOP_ LOCA) Now, we can apply binary logistic regression. APC data of 7 bus stops of route 1 were used to develop the model. At each stop, the sums of passengers through the front door and the rear door were calculated for each door and compared to each other for every bus trip. If the front door is the busier door then it is coded as 1, otherwise as 0. In this research, the dwell times longer than 180 seconds or equal to zero are not included in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 132 the model. Also, the cases that recorded boarding passengers at the rear door are considered as “unusual events” and erased. The validated dataset is shown in Table 5.5. Out of the 2642 cases, there are 1740 cases recording that the busier door is the front door, accounting for 68.3 percent of the total. The backward stepwise method was used in this research in order to delete the variables with significance level larger than 0 .1, and as the sequence, the best regression was being found. To do this, the SPSS 13 package was used to develop the model for each type of U presented in Equations 5.41, 5.42 and 5.43. The overall significance was tested with Model Chi-squared as called by SPSS, which is derived from the likelihood of observing the actual data under assumption that the model that has been fitted is accurate. The best regression function of each type from B-L.l to B-L.3 after the backward stepwise procedure were found and tabulated in Table 5.6. Details can be seen in Appendix B2. Table 5.5: Validated Dataset for Logistic Regressions VALIDATED BUSIER DOOR Cumulative Frequency Percent Valid Percent Percent Valid .00 808 30.6 31.7 31.7 1.00 1740 65.9 68.3 100.0 Total 2548 96.4 100.0 Missing System 94 3.6 Total 2642 100.0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 Table 5.6: Best Binary Logistic Regression Equations -2LL Nagelkerke Overall Type The Best Logistic Regression Equation R Accuracy quare U = -1.023+ i.205(TOTAL ONS)-0.3\4(TOTAL OFFS) + =562.67 55^=0.00 Sig=0.00 Sfg=0.00 B-L.l 1804.39 0.586 86.3 % 0.0\1(LOAD ARR) - 0.001 .(PUNT)+ 0.104 .(STOP LOCATE( 1)) St* =0.002 — Sig=0.073 S/g=0.000 — B-L.2 U = 0.510+ 1.091 .(TOTAL O N )- 0.018 .(TOTAL O N)2 948.378 0.469 89% / 2 =2520.600 Sig =0.063 Sig =0.000 S * =0.000 - 0.028 (LOAD ARR) - 0.821 (STOP LOCA ) Sig =0 000 — =0.001 ~ U = 1.227 + 1,460(TOTAL O N ) - 0.665 (TOTAL O FF) - Sfc =0.00 0.00 Sig -0.000 0.024 .(TOTAL O N )2 + 0.018 (TOTAL OFFS)2 - Sig =0.000 — Sig =0.000 — B-L.3 861.847 0.529 89% 0.022 (TOTAL ONS \TOTAL OFFS) - 0.042 .(LOAD ARR) Sig =0.000 — " % =0.000 — As can be seen in Table 5.6, all types are quite powerful in term of prediction. To obtain further information for selection, we use the R2^ which was suggested by Menard (2001, p.27). According to him “R2l is the most appropriate for logistic regression because it is conceptually closest to the Ordinary Least Square Rr = (5.44) X-2LL Where: z 2 = the Chi-square of the logistic regression model. Following Equation 5.44, the R2L for types B-L.l, B-L.2 and B-L.3 are 0.433, 0.372 and 0.430, respectively. Type B-L.l shows the highest R2l■ As a result, it is used for this research. Table 5.7: Model Summary of Type B-Ll -2 Log Cox & Snell Nagelkerke R Step likelihood R Square Square 1 1804.229(a) .418 .586 2 1804.777(a) .418 .586 a Estimation terminated at iteration number 8 because parameter estimates changed by less than .001. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 Table 5.8: Prediction Performance of Type B-L.l Classification Table? Predicted BUSIER DOOR Busier door Busier Door Percentage Observed is rear door is Front door Correct Step 1 BUSIER_DOOR Busier door is rear door 647 161 80.1 Busier Door is Front door 189 1551 89.1 Overall Percentage 86.3 Step 2 BUSIER_DOOR Busier door is rear door 647 161 80.1 Busier Door is Front door 188 1552 89.2 Overall Percentage 86.3 a- The cut value is .500 As shown in Table 5.7, it has a pseudo R 2 statistics of 0.586 (Nagelkerke R2) and 0.418 (Cox & Snell R2), which indicate a satisfactory model. Table 5.8 presents prediction performance of type BL-1. The overall correct prediction of the model is up to 86.3 percent. Out of the 1740 cases recorded showing that the busier door is the front door, this model predicted correctly 1552 cases. This results in an accuracy of up to 89.1 percent. Also, out of the observed 708 cases where the rear door is the busier door, the model predicted correctly 647 cases, resulting in a good prediction up to 80.1% of accuracy. Table 5.9 tabulates the coefficients of type B-L.l. The odds ratio associated with each coefficient is presented as Exp(B) shown in the last column of this table. The odds ratio is the number by which we would multiply the odds fr 0nt_door (the probability divided by 1 minus the probability, as shown in Equation 5.34) for each one-unit increase in the dependent variables. An odds ratio greater than 1 indicates that the oddsfront_door Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. increases when the independent variable increases and an odds ratio smaller than 1 indicate that the oddsfront joor decrease when the independent variable increases. Table 5.9: Coefficients of TypeBL-1 B S.E. Wald df Sig. Exp(B) Step TOTAL_ONS 1.203 .059 419.818 1 .000 3.331 1(a) TOTAL_OFFS -.314 .021 217.103 1 .000 .730 LOADARR .017 .006 9.586 1 .002 1.018 l PUNT © o .000 3.190 1 .074 .999 SEASON 7.004 .072 SEASON(1) -.183 .154 1.416 1 .234 .833 SEASON(2) -.088 .172 .260 1 .610 .916 SEASON(3) .238 .170 1.954 1 .162 1.268 STOP_LOCATE(1) .704 .177 15.838 1 .000 2.023 TIME .548 .760 TIME(1) .089 .156 .328 1 .567 1.093 TIME(2) .112 .154 .526 1 .468 1.118 Constant -1.105 .250 19.548 1 .000 .331 Step TOTALONS 1.205 .059 420.852 1 .000 3.337 2(a) TOTAL_OFFS -.314 .021 218.743 1 .000 .730 LOADARR .017 .006 9.620 1 .002 1.018 1 PUNT o o .000 3.213 1 .073 .999 SEASON 6.913 .075 SEASON(1) -.186 .154 1.462 1 .227 .830 SEASON(2) -.096 .171 .317 1 .573 .908 SEASON(3) .230 .170 1.840 1 .175 1.259 ST OP_LOCATE( 1) .704 .177 15.844 1 .000 2.023 Constant -1.023 .223 21.020 1 .000 .359 a Variabte Because the odd ratios of the independent variables TOTAL ON, LOAD_ARR and STOPLOCATE(l) are all greater than 1, the oddsfrontjoor will increase when these variables increase. In contrast, the oddsfronl joor will decrease with the increments of the variables TOTAL OFF and PUNT. Specifically, while controlling other variables, if the bus has one more boarding passenger, the oddsfrontj oor will increase 3.33 times. This is obvious because all Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 passengers have to board at the front door. Similarly, the oddsfro„tjoor will increase 1.02 with one unit loading factor increment. This result is expected because on-board passengers tend to alight via front door because of the difficulties in circulation. This odds will be doubled if the bus serves in CBD area. In contrast, if the bus has one more alighting passenger, the oddsfrontj oor will decrease 0.83 times. Also, the oddsfrontjoor will reduce if the bus arrives at stop later. Once the busier door has been defined, dwell time will be determined by Equations 5.45, 5.46 and 5.47. dwell =P{.(TOTAL_OFFS)la +{TOTAL ONS)Ib+top if Pj>P2 (5.45) dwell= Max { P{ .(TOTAL _ OFFS)Ja + (TOTAL _ ONS).tb; P2.( TOTAL_OFFS).Q + t op if P, P 2= 1-P, (5.47) Where: TOTAL ONS and TOTAL OFFS = Total the number of boarding and alighting passengers. ta, tb = Average service time per passenger for alighting and boarding, respectively. top = Door opening and closing time ( 4 seconds) 5.2.6.2 Articulated Bus Unlike a rigid bus, an articulated bus has three doors and passengers can board via rear doors. To predict the busiest door out of the three doors of the bus, multinomial Logit regression is applied. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 Mathematically, the extension of the binary logistic regression, which was applied for rigid-body bus, to multinomial logistic regression is straightforward. Let us assume the second rear door is the reference category (i.e., the probability that this door is the busiest door). We call the natural logarithm of the odd (jront door) (the ratio of the probability that the front door is the busiest door to the probability that the second rear door is the busiest door) is the Logit of front door, written as Logit (front door). Similarly, we denote Logit (first_rear door) as the natural logarithm of the ratio of the probability that the first rear door is the busiest door to the probability that second rear door is the busiest door). In mathematical notations, we have: Logit(front door) = In (5.48) p Logit( first _ rear _ door) = In — (5.49) P 1+P2+P3-I (5.50) Where: Pi, P2, and P3 = the probabilities that the front door, the first rear door and the last rear door is the busiest door, respectively. In = natural logarithm or logarithm base e Similar to the explanations in the case of two-door bus, we use the convenience functions namely Uj and U2 whose variables are collectable from APC and AVL systems, one for each door (i.e., the front door and the first rear door) relative to the second rear door (i.e., the reference category), to describe the relationship between the probabilities and the collected data. In other words, we model Logit (front door) = Uj and Logit Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 (first_reardoor) = U 2. Therefore, based on Equations 5.48, 5.49, and 5.50 with several mathematical operations, we have the probability that the front door is the busiest door is: e u' Pi= ~0eu'---- +eUl+lu 7 <5'51) Similarly, the probability that the first rear door is the busiest door is: P2 = ~ d --- d — 7 (5.52) e ' + e 2 +1 And, the probability that the second rear door is the busiest door is: <5 -5 3 > Some types of Uj and U2 equations are examined in order to find the best multinomial Logit regressions. The types of U ’s function are formulated as follows Type B-ML.1 Ul=P0+ .(TOTAL _ ONS) + /?2_, .(TOTAL _ OFFS) + /?3_, .(LOAD _ ARR) + /?4_, .(LF) + (5 .5 4 ) + y?51 .(PUNT) + /V , .(TIME) + /?7_, .(SEASON)+ /?8_, .(STOP _ LOCA ) U 2 = pa + P^2 .(TOTAL _ ONS) + 0 2_2 .(TOTAL _ OFFS) + P,_2.(LOAD _ ARR) + 0 ^ 2.(LF) + (5.55) + /?J2.(PUNT)+06_2 .(TIME) + P2_2 .(SEASON) + 0%2 .(STOP _ LOCA) Type B - M L. 2 U, = 0 (l + 0 I_I.(TOTAL_ONS) + 0 1_I.(TOTAL OFFS) + fl.gTOTAL _ O N f + p^.(LOADARR) + (5.56) .(LF) + +/?6_, .(PUNT) + 0 2_,.(TIME) + .(SEASON) + .(STOP _ LOCA) U2 = 0O + 0 ^ V(TOTAL ONS) + P2_2.(TOTAL OFFS) + 0,_2(TOTAL ON)2 + (5.57) 0 4.2.(LOAD _ ARR) + 0s_2.(LF) + +06_2.(PUNT) + 01_1.(TIME) + 0t _2.(SEASON) + 09_2.(STOP _LOCA) Type B - M L. 3 Ut ~ Po + Pi-, .(TOTAL _ O N S ) + 0 2_,.(TOTAL _ OFFS ) + 0 ,^ (TOTAL _ ON ) 2 + 0 4_, (TOTAL _ O F F )2 (5.58) + (TOTAL _ ONS )(TOTAL _ OFFS ) + 0 t_, .(LOAD _ ARR ) + 0 ,_ ,.(L F ) + + 0 s_t.(PUNT ) + .(TIME ) + .(SEASON ) + 0„_,.(STOP _L O C A ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 In order to find the coefficients of the regression types above, APC data from 7 stops on the bus route 95 were used. Total number of boarding and alighting passengers at each door of the bus at each bus stop were calculated and compared with those of the other doors. If a door is the busiest door, it is coded as 1 and the remaining doors are coded as zeros. By using the multinomial logistic regression function of the SPSS 13.0 software for each type of Us named as B-ML.l, B-ML.2 and B-ML.3, we have the best Multinomial Logistic Regression for each. The results of these procedures are tabulated in Table 5.10. Please see Appendix B3 for the detail of each type. Table 5.10: The Best Multinomial Logistic Regression Equations McFaden x 2 - R Overall Type The Best Logistic Regression Equation Accuracy deviance sq u a re {/, = 1.302 + 0.098 •TOTAL ONS - 0.308 • TOTAL _ OFFS - 6.10 ~3 * LOAD ARR -0 .1 0 5 • TIME + 6 .1. 10 * SEASON + 0 .549 * STOP _ LOCA B- 10360.49 0.193 62.3 M L.l U2 = -0.133 - 0.001 * TOTALONS-0.023* TOTAL_OFFS+ 0.004* LOADARR-0.114* TIME- 0.022* SEASON+ 0.057* STOP_LOCA 17, =1.602 + 1.50 * TOTAL _O N S - 0.30 * TOTAL _OFFS - 0.003 * TOTAL _ O N 2 - 0.013 * LOAD _ ARR - 0.116 * TIME + 0.122 * SEASON + 0.659 * STOP _LOCA B- 8352.10 0.194 65.2 ML.2 U 2 = -0.228 + O.Ol6(TOTAL _CWS) - 0.023 .(TOTAL _OFFS) + 0.003LOAD _ ARR ) - 0.104.(77A4E) - 0.011(SE4SCW) + 0.034.{STOP _ LOCA) Ut = 2.230+ 0.166* TOTALJONS- 0.513* TOTALOFFS- 0.006* TOTAL_ON2 + 0.006* TOTAL_OFF* + 0.01 * TOTAL ONS* TOTAL_OFFS - 0.016* LOAD_ARR- 0.172* TIME+ 0.133* SEASON+ 0.601* STOP_LOCA B- 861.85 0.224 65.9 ML.3 C/2 =-0.178+0.19* TOTAL_ONS- 0.032 * TOTALOFFS- 0.001* TOTALOFt +0003* LOAD A R R -1.05* TIME- 0.01 * SEASON^ 0.033* STOP_LOCA Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 As shown in Table 5.10, type ML-3 provides the largest McFaddden R2l of 0.224 and the overall accuracy of 65.9 percent. Therefore, type ML-3 is suggested for busiest door prediction of articulated bus. Also, it works fairly well, as indicated by x 2 °f 861.847 and McFaddden R2l of 0.224. The overall Likelihood Ratio tests of this type are presented in Table 5.11 on the basis of SPSS application. The results show that all parameters are statistically significant, except the variable PUNT. Table 5.11: Likelihood Ratios of type B-ML.3 Model Fitting Criteria Likelihood Ratio Tests Effect -2 Log A lC of BICof Likelihood of Reduced Reduced Chi-Square df Sig. Reduced Model Model Model Intercept 8372.224 8503.287 8332.224 285.443 2 .000 LOAD_ARR 8090.781 8234.950 8046.781(a) .000 0 LF 8090.781 8234.950 8046.781(a) .000 0 PUNT 8086.832 8217.895 8046.832 .051 2 .975 SEASON 8104.913 8235.976 8064.913 18.132 2 .000 STOP_LOCATE 8143.819 8274.882 8103.819 57.038 2 .000 TIME 8099.926 8230.989 8059.926 13.145 2 .001 TOTAL_ONS 8213.854 8344.917 8173.854 127.073 2 .000 TOTAL_OFFS 9061.769 9192.832 9021.769 974.988 2 .000 Total_On_squared 8205.349 8336.412 8165.349 118.568 2 .000 TotalOffPw 8125.214 8256.277 8085.214 38.433 2 .000 TOTALon_and_OFF 8191.669 8322.732 8151.669 104.888 2 .000 The chi-square statistic is the difference in -2 log-likelihoods between the final model and a reduced model. The reduced model is formed by omitting an effect from the final model. The null hypothesis is that all parameters of that effect are 0. a This reduced model is equivalent to the final model because omitting the effect does not increase the degrees of freedom. As shown in Table 5.12, the probabilities of front door and the first rear door are both reduced when more passengers get off the buses, but the front door has a larger reduction. In contrast, these probabilities increase when more passengers get on the bus Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 and the influence on the front door has a larger increment. Interestingly, location of bus stop has a quite strong influence on passenger’s door choice. This can be seen shown in Logit (door_1 busiest) and Logit (door_2 busiest) of 1.824 and 1.033, if the stop is located in the CBD area. Table 5.12: Parameters of Type B-ML.3 Parameter Estimates 95% Confidence Interval for Exp(B) Busiest dodt B Std. Error Wald df Siq. Exp(B) Lower Bound Upper Bound door 2 busiest Intercept -.178 .179 .993 1 .319 LOAD_ARR .003 .003 1.300 1 .254 1.003 .998 1.008 LF 0b PUNT .000 .000 .008 1 .930 1.000 .999 1.001 SEASON -.010 .039 .061 1 .806 .990 .918 1.069 ST OP_LOCATE .033 .092 .127 1 .721 1.033 .863 1.237 TIME -.105 .051 4.232 1 .040 .901 .815 .995 TOTAL_ONS .019 .017 1.276 1 .259 1.019 .986 1.054 TOTAL_OFFS -.032 .014 5.113 1 .024 .969 .942 .996 Total_On_squared -.001 .001 1.115 1 .291 .999 .998 1.000 TotalOffPw .000 .000 .621 1 .431 1.000 1.000 1.001 TOTALon_and_OF .000 .001 .112 1 .738 1.000 .998 1.001 Door 1 busiest Intercept 2.230 .159 195.865 1 .000 LOAD_ARR -.016 .002 43.579 1 .000 .984 .979 .989 LF 0b PUNT .000 .000 .022 1 .882 1.000 .999 1.001 SEASON .133 .036 13.534 1 .000 1.142 1.064 1.226 STOP_LOCATE .601 .088 47.085 1 .000 1.824 1.536 2.166 TIME -.172 .048 12.841 1 .000 .842 .767 .925 TOTAL_ONS .166 .016 102.289 1 .000 1.181 1.143 1.219 TOTAL_OFFS -.513 .018 851.036 1 .000 .599 .578 .620 Total_On_squared -.006 .001 92.083 1 .000 .994 .993 .995 TotalOffPw .006 .001 53.559 1 .000 1.006 1.004 1.007 TOTALon_and_OF .010 .001 72.065 1 .000 1.010 1.007 1.012 a. The reference category is: door 3 busiest. b- This parameter is set to zero because it is redundant. Following the best multinomial logistic regression type, we can estimate the dwell time at the busiest door, as shown in Equations 5.60 and 5.61. P Busiest_ door Max (Pi, P2, and P3) (5.61) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 Pi, P2, P3 are the probabilities that the busiest door is the front door, the first rear door and the second rear door defined in Equations 5.51 to 5.53 and type B-ML.3, respectively. 5.2.7 Method A vs. Method B and the Selection Two methods (A& B) and their components have been developed to predict dwell time in the previous sections. Method A is based on the best dwell time regression function (i.e., type A-2.3) to examine the relationship between dwell time and the number of boarders and alighters as well as other variables. Method B is based on the revisions on busiest door selection method. Once the busiest door is selected on the basis of probability, dwell time is predicted by using the predicted number of boarders and alighters at that door as shown in Equations 5.45 and 5.46 for rigid-body buses or Equations 5.60 and 5.61 for articulated buses. In order to find enhanced methodology for use in this research, two methods are compared, based on their prediction performance in terms of the overall accuracy. Overall accuracy can be determined by calculating one of the measures presented in Table 5.13. In this study, the mean absolute percentage error (MAPE) is used as a yardstick to select the better of the two methods. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 143 Table 5.13: Measures of Accuracy Measure Equation MAE Mean Absolute MAE = xi ~ x ‘ 1 (5.62) f A ) MAPE 1 » |v _ v | Mean Absolute Percentage MAPE= Y 1 ‘ 1 .100 (5.63) Error N m xt V MRE Maximum Relative Error MRE = max— — — (5.64) x, RMSE Root Mean Square Error RMSE = ^ f i(xl-xl)2 (5.65) 5.2.7.1 Method A In method A, as shown in type A-2.3, dwell times are predicted as below: dwell = 14.434+ 2.014(TOTAL O N )- 0.026 .(TOTAL ON)2 +0.5\6(TOTAL OFFS) + /=13.921 f=54.071 f=-17.944 f=15.364 O.OOMTOTAL OFFS)1 - 0.022 (TOTAL ONS)(TOTAL OFFS) + 0.078 .(LOAD ARR) f=3.387 — t = - \ 1.548 — ~ 2.982 - 5.047(ZF) + 0.003(PUNT) - 4.748 (DOORS) + 2M7(STOP LOCA) + 0.196 (SEASON) -3.349 (=5.492 (=-12.295 (=10.384 — 1=2.419 S.2.7.2 Method B While method A predicts dwell time without clustering data, method B clusters data into two samples, one for rigid-body bus and the other for articulated bus. The values ta, tb, are in accordance with the HCM (2000) suggestions (Exhibit 27- 9, p.27-10) as shown in Table 5.14. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 5.14: Average Service Time Selection(s/p) Average dwell time per boarding Average dwell time per alighting Bus Type passenger through passenger through Front door Rear door Front Door Rear Door Rigid-Body Bus 1.2 NA 0.9 0.9 Articulated Bus 0.9 0.6 0.6 0.6 5.2.13 Rigid-Body Bus In the case of rigid-body bus, dwell times are predicted by using Equations 5.45 to 5.47 and the logistic regression type namely B-L.l. Table 5.15 shows a fraction extracted from the outputs of two dwell time prediction methods applied for rigid-body buses running on bus route 1, the OC Transpo, Canada. 5.2.1 A Articulated Bus In case of articulated bus, dwell times are predicted by using Equations 5.60 and 5.61 and the logistic regression type namely B-ML3 whose multinomial logistic regression functions are: Ux = 2.230 + 0.166 * TOTAL _ ONS - 0.513 * TOTAL _ OFFS - 0.006 * TOTAL _O N 2 + (5. 67) 0.006 * TOTAL _ OFF2 + 0.01 * TOTAL _ ONS * TOTAL _ OFFS - 0.016 * LOAD _ ARR - 0.172 * TIME + 0.133 * SEASON + 0.601 * STOP LOCA C2 =-0.178 + 0.19* TOTAL _ ONS - 0.032. * TOTAL _OFFS - 0.001 * TOTAL _ O N 2 (5.68) + 0003 *LOAD ARR -1.05 * TIME - 0.01 * SEASON + 0.033 * STOP LOCA Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 145 Table 5.15: Example Output of Dwell time Prediction of Method A and Method B Applied for Rigid-body-Bus Predicted_ Actual_ Predicted_ Bus Actual_ Arr_ dwell_ ExpecTime Date Dwell dwell_ Stop Time Method A _ Time (s) Method B(s) (s) CH080 33105 32880 9/14/2004 7 11.88 6.07 CF630 32700 32580 9/14/2004 6 CD920 33840 33600 9/16/2004 19 27.84 24.03 RA190 30375 30540 9/24/2004 2 4.39 2.65 CC190 29220 29280 10/5/2004 42 33.67 41.6 RF075 31500 31500 10/6/2004 5 7.29 2.46 CC190 33570 32880 10/8/2004 7 10.04 6.35 CC190 30510 30480 10/8/2004 28 18.49 22.69 CD920 33240 32400 10/8/2004 15 26.3 18.29 CF630 29085 28980 10/8/2004 4 10.07 4.21 CH080 29520 29280 10/8/2004 9 8.34 4.8 CH080 32490 31680 10/8/2004 12 15.26 7.59 CD920 30330 30000 10/14/200 4 9.37 4.2 CH080 29595 29280 10/14/2004 9 8.95 3.99 CH080 29460 29280 10/18/2004 10 13.61 8.63 CD920 32850 32400 10/20/2004 30 32.03 23 RA190 31680 31740 10/20/2004 4 5.48 3.46 RA945 31140 31080 10/20/2004 12 8.71 3.95 RA945 31110 31080 10/20/2004 10 16.14 9.19 CH080 31980 31680 10/20/2004 11 15.43 9.42 RA190 30480 30540 10/25/2004 10 11.73 7.59 RA945 31125 31080 10/25/2004 31 32.88 25.4 CF630 31605 31380 10/25/2004 7 10.62 4.94 RA945 34635 34680 10/27/2004 5 6.42 2.71 RA945 34560 34680 10/27/2004 25 23.97 16.1 CF630 35025 35100 10/27/2004 2 8.08 2.93 CD920 33690 33600 11/1/2004 13 17.52 10.74 CH080 33105 32880 11/1/2004 7 10.87 4.69 Table 5.16 shows another fraction of the outputs of the two dwell time prediction methods applied for articulated buses running on bus route 95, the OC Transpo, Canada. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 146 Table 5.16: Example Output of Dwell time Prediction of Method A and Method B Applied for Articulated Bus Predicted Predicted Actual_ Bus _dwell_ _dwell_ A c tu a lA rrT ime ExpecTime Date Dwell_ Stop Method A Method B Time (s) (s) s) EE915 29475 29400 9/7/2004 19 23.41 7.73 CJ900 30975 30720 9/7/2004 9 9.91 4.9 CA920 30780 30420 9/7/2004 7 14.67 6.64 CA920 31620 31440 9/8/2004 9 10.61 5.38 CJ900 31830 31740 9/8/2004 16 7.07 3.65 NI910 32415 32340 9/8/2004 9 5.04 3.56 EB905 30615 30600 9/8/2004 16 20.05 6.95 CD910 31245 31140 9/8/2004 3 6.7 3.86 CJ900 33345 33300 9/9/2004 4 4.6 3.23 EB905 32250 32160 9/9/2004 8 15.51 8.23 CD910 32775 32700 9/9/2004 24 18.24 7.31 EE915 35730 35820 9/10/2004 26 16.07 5.86 EE915 31575 31500 9/10/2004 21 11.69 5.89 EB905 31785 31680 9/10/2004 7 7.83 4.62 CA920 32730 32520 9/10/2004 8 11.41 7.12 CA920 30480 30240 9/10/2004 11 11.02 4.41 NI910 32505 32340 9/10/2004 2 3.27 2.8 NI910 32490 32340 9/10/2004 3 2.28 2.86 NI910 31410 31080 9/10/2004 3 4.26 3.25 CE940 32100 31980 9/10/2004 3 3.14 2.74 EB905 30585 30600 9/10/2004 5 8.71 5.81 NI910 35940 35940 9/14/2004 2 2.92 3.1 CJ900 35400 35340 9/14/2004 1 3.57 2.52 EE915 31575 31500 9/15/2004 17 28.93 11.17 EB905 31770 31680 9/15/2004 8 13.07 4.73 CE940 32115 31980 9/15/2004 12 15.65 8.11 CJ900 32100 32100 9/17/2004 25 9.14 4.93 NI910 32655 32700 9/17/2004 6 5.53 4.27 CD910 31575 31500 9/17/2004 21 20.94 5.94 CD910 33255 32880 9/17/2004 11 13.09 8.34 CJ900 33585 33420 9/20/2004 2 3.88 2.53 5.2.7.5 Accuracy Performance of the Two Methods Obviously, the higher the value of MAPE, the poorer the prediction is. We can see from Table 5.17 that while method A is quite poor in predictions for rigid bus (i.e. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 MAPE up to 59.80%) it works well for predicting dwell time for articulated buses with MAPE as low as 35.79%. In contrast, method B outperforms method A in case of rigid- body bus (MAPE= 32.42%) as compared to that of articulated bus (MAPE=45.72%). Table 5.17: Mean Absolute Prediction Error of Method A vs. Method B MAPE (%) 20% sample size Bus Type Method A Method B Rigid-body-Bus 59.80 32.42 473 cases Articulated Bus 35.79 45.72 1846 cases As a result, method A is suggested for predicting dwell time for an articulated bus while method B should be used for a rigid-body bus. 5.3 Summary A comprehensive research methodology and proof of concept analyses on dwell time predictions are reported in this chapter. Dwell time in previous works found in the literature was usually considered as delay and its values were defined by multiple regressions where the most influencing variables are total alighters and boarders. When using the models developed in the past to predict dwell time at a stop, the analyst is asked to enter the number of passengers boarding and alighting at that stop. Unfortunately, these numbers are not available because the bus has yet come to the stop. The dwell time prediction model developed in this chapter is a dynamic model because it can predict the number of boarding and alighting passengers. Based on the statistical pattern recognition technique, the historical Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 148 data and the up-to-minute data transmission from the APC and AVL systems, the predictions of passengers’ activities are predicted and updated dynamically by the two sub-modules, which were called the RBSM and the RASM. Method A and method B were developed to examine the fact that dwell time has a complicated relationship with many characteristics of passenger activities and service activities. A series of regression types varying from multiple linear regression, namely non-linear regression, binary Logit regression, and multinomial Logistic regressions were applied to explore this relationship. Data collected by the APC system mounted on the OC Transpo buses were used as the inputs to these methods. The comparisons among them show that method A (i.e. non-linear regression model) outperformed method B (i.e., Multinomial Logit regression) in predicting dwell time for the articulated bus. On the other hand, method B (i.e. binary Logit) is suitable for the rigid-body bus. The performance of boarding passenger prediction and alighting passenger prediction sub-modules is presented in the next chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 MODEL PERFORMANCE 6.1 Introduction In the previous chapters, the study framework, the methodologies, the mathematical algorithms, and the detailed procedures for each constituent element of the developed model have been presented. Prior to application, it should be evaluated in terms of accuracy and reliability. From the accuracy perspective, the model’s predictions on bus arrival time should be evaluated on the basis of several measurements of goodness-of-fit. Furthermore, these predictions would also be compared with predictions resulting from the use of other predictors, given the same input data. In terms of reliability, the developed model should be tested with different bus running scenarios in order to assess if it is a versatile model and if its predictions are reliable under any scenario. This chapter presents information on the conversion of the developed mathematical prediction algorithms into computer programs, simulation of several bus operational scenarios, and evaluation of the developed model’s performance based on simulated as well as real-world data. 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 150 6.2 Computer Program Development The developed model was coded in Matlab computer language, version 7.0. The computer program consists of two main parts, called “functions” by Matlab. Each function contains several sub-functions in order to make the whole program work efficiently. The structure of the computer program, its functions, and sub-functions are shown in Figure 6.1. The source codes for each can be seen in Appendix C of this document. Function: Bus_running_dme Function: Bus dwell time Sub-function: Sub-function: deboarders boarders Figure 6.1: Structure and Constituent Functions of the Computer Program 6.3 Simulations of Bus Operation Scenarios A number of bus operation scenarios that are difficult to carry out as field experiments were simulated. The simulated bus arrival times, dwell times, and passenger activities were used to test the developed model in term of its accuracy and reliability. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 Three bus operation scenarios were set up. These are: bus running under normal traffic condition, bus running under incident condition, and bus running under high passenger demand at some busy stops. It should be noted that the purpose of these simulations is to experience randomness of real-life bus operations generated by simulation tools. This approach goes beyond the analysis of a single operation case. In order to obtain simulated data, transit routes were coded in suitable traffic software which was calibrated before being used for simulation purposes. 63.1 VISSIM Simulator VIS SIM is a popular microscopic traffic simulation software, Verkerhr In Stadten-SIMulation (PTV 2004). Like other traffic microsimulators such as CORSIM, SIMtraffic, etc., a key feature of VISSIM is the capability of simulating the randomness of traffic flow. The vehicle-following, lane-changing, and route selection components of VISSIM are recognized to mimic the real-world traffic operations. Furthermore, the flexibility and convenience of modelling interactions such as stopping, yielding, queuing, and passing, as well as the plentiful user-defined parameters, all make VISSIM to be a very useful traffic simulation tool. VISSIM was chosen for this study due to not only attributes noted earlier but also owing to capability to simulate transit. Other simulators have limitations in this aspect. With VISSIM, a public transit planner can define different bus routes for the same road; each with corresponding time-based schedule or headway-based schedule. At each bus stop, passenger activities such as dwell time, number of boarders and de-boarders can be defined separately following either normal distribution or empirical distributions for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 every single bus route sharing the same bus stop. Furthermore, VISSIM allows the users to define various types of acceleration and deceleration characteristics of the buses under study as well as bus driver behaviour. These enable transit planners and operators to predict effects of locations of transit facilities and passenger activities, generation of the real-time traveller information, and the study of operational alternatives before actual implementation. Table 6.1 compares some transit-related features of VISSIM versus CORSIM, the other commonly used micro-simulation software. Table 6.1: Comparison of VISSIM (version 4.10) and CORSIM (TSIS 5.1) for public transit simulation Module VISSIM CORSIM Dispatching buses - Headway-based schedule 1 - Headway-based schedule - Time-based schedule 2 - N /A 6 Distribution of dwell time - Normal Distribution -Mean dwell time and - Empirical distribution distribution tables. Estimate dwell time - Additive method 4 - Additive method - Maximum method 5 -N/A Estimate number of - Number of boarders and -N/A boarders and de-boarders de-boarders Estimate number of waiting - Capability exists -N/A passenger Calculate passenger waiting - Capability exists -N/A time Number of dedicated bus - 2 or more - only one lanes Notes 1 Headway-based schedule: Users can define headway (or service frequency) and start trip number. 2' Time-based schedule: Users can define the time the bus is dispatched at all bus terminals. 3 Empirical distribution: Users can define empirical distribution of dwell time at any bus stop for any bus route based on actual dwell time. 4 Additive method: Dwell time is calculated by additional boarding and alighting times. 5 Maximum method: Dwell time is calculated based on the maximum service time for boarding or alighting passengers, otherwise sum of both. This is suitable for one directional door. 6 N /A : Not applicable Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 153 6.3.2 Transit Route Coding in VISSIM The two bus route segments of interest in this research, namely route 95 and route 1, were coded in VISSIM. Traffic and geometric data for each were provided by the City of Ottawa while the data on bus operations were obtained from APC systems of the OC Transpo. Several field surveys were also made at sites for which data were not available. A bitmap file of the routes’ layout was imported into the simulator and the network geometry was built based on the background map using links and connectors. The traffic network coding required much effort because VISSIM specifies elaborate coding procedures for every single facility of the network. Other supplemental features such as number of lanes, speed limits, stop signs, right-tum-on-red, and actuated traffic signals, were also carefully coded. Furthermore, traffic and signal timings for intersections for the 2-hour peak period in the morning (8:00 a.m.-10:00 a.m.) were coded. Since operation of the buses on selected routes was of much interest in this research, the kinetic characteristics of the actual buses (i.e., acceleration and deceleration rates) running on the OC Transpo network reported by Zargari (1997) and Zargari and Khan (2003) were used (see Table 6.2) instead of using VISSIM default parameters. Table 6.2: Kinetic Characteristics of the OC bus Parameter Maximum Acceleration Maximum Deceleration (m/s2) (m/s2) VISSIM defaults 1.52 4.58 Zargari and Khan (2003) 1.72 4.02 In order to make the simulated bus operation to be more realistic, besides the selected bus stops for which the actual APC data on passenger activities (e.g., the number Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 154 of boarders and alighters) were used as the VISSIM inputs, other supplemental bus stops were also included in the routes. For these stops, empirical distributions of dwell time at bus stops were used (e.g., mean, standard deviation). Also, real-world bus dispatching schedules were imported for each route. The two bus routes in the coded form are shown in Figure 6.2. VISS1W4.1G 13 •c:\myoTt.wa\oc.tfdnspo.inp ' CfSlixi B m M a TWIfc S^tal Cortrol * 0 - -v. w»! bJbl U IWi U Figure 6.2: Bus Routes 95 and 1 coded in VISSIM 4.1 VISSIM can automatically record the history of every single bus running on the bus route at time steps as small as 0.1 second interval and tabulates the data into the built- in table, namely the Vehicle Record. The software can provide over 75 fields in a such a table varying from kinetic information of the vehicles to emissions, fuel consumption, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 155 passenger activities of public transit (i.e., dwell time, the number of boarders and de boarders, passenger waiting time, bus arrival time and departure time, etc.,), vehicle delay (e.g., queuing, number of stops, etc.,), and GPS data (e.g., X, Y, Z coordinates). Figure 6.3 shows the user-defined process in VISSIM for selecting transit bus parameters of the simulated network. Vehicle Record - Configuration [Xj Selected param eters Parameter selection Start Time HRi i— i j Qe|ay Time PT: Lne Number ; ( J ? | '• Desired headway PT: Course Number Desired Lane PT: Transit stop number ' D esired Speed [km/h) 1 PT- Boarding P assen g ers • Desired Speed [m/s] Up Down T ine [s] Configuration file: jo ctran sp o fzk (77) □ h d u d in g parked vehicles 0 Database Table name: octranspo_VEH_RECORD |5ct Jliing Wl, OK Cancel Figure 6.3 Vehicle Record-Configuration Tool in VISSIM 6.3.3 Calibration and Validation Micro-simulation is useful but dangerous if it is not carried out properly (Fox, 2000). Although the purpose of simulations in this study is to take advantage of randomness of transportation data generated by VISSIM for the developed model testing, in order to enhance its value, the calibration is needed. The calibration process applied in this study follows the instructions provided by Dowling et al., (2004). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 156 The process started with on-screen animation so that coding errors could be easily detected. Each time an observed vehicle’s behaviour appeared to be unrealistic such as unusual stop on the route, left turning without priority rule, unrealistic collision, or unrealistic congestion, the potential causes were analyzed and the relevant parameters were adjusted. This process was repeated several times in order to ensure that the simulated network is consistent, the coding is accurate, and the simulation is a reality. Once the on-screen error checking was completed, the calibration process progressed to the next step. In doing so, the two most influencing parameters which have a major influence on safety distance between vehicles and thus affect the saturation flow rate were chosen for calibration. These parameters are BXADD and BXMULT, where BXADD is the additive part and BX MULT is the multiplicity part of desired safety distance (VISSIM 4.1 User Manual, p.5-26). Please refer to Wiedemann et ah, (1994) for further details. Based on these parameters, the calibration targets were set up. Bus running times between a pair of stops (i.e., the links) generated by VISSIM were compared with actual running times of the OC Transpo buses. According to the calibration targets, the differences should be less than 15 percent for each link, and the number of such links should account for more than 85 percent of the total. These are suggested by Dowling et al., (2002, p.64). Details of the calibration process and results are presented in Appendix Dl. Table 6.3 shows the calibrated values that were used for the follow-up steps. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 157 Table 6.3: VISSIM Calibrated Parameters Default Calibrated Parameters values values Average stand still 2.0 1.6 distance (m) BX ADD (m) 3.0 2.0 BX MULT (m) 3.5 3.0 Gap acceptance (s) 5 7-10 1 Note: 1 The calibrated value was based on the on-screen error checking 6.3.4 Bus Operation Scenarios, Micro-simulation Runs, and the VISSIM Outputs The simulation model for the two bus routes and part of traffic network was coded and calibrated in VISSIM. This enabled the use of VISSIM to generate data on bus operations and passenger activities that correspond with transit and traffic conditions. Three scenarios were simulated. The first scenario related to routine traffic, and the hourly and daily traffic variations were simulated. The second scenario involved an incident on the bus route. The incident could be a one-lane closure or a slowdown zone for road maintenance, or it may be due to traffic accident. The last scenario was devoted to the simulations of passenger activities at some busy stops that were caused by special events such as Boxing Days, or a football match. The number of simulation runs was also carefully considered. In VISSIM, each simulation run requires a random seed number, which is used by VISSIM to generate traffic volumes and vehicle arrival rates. Consequently, multiple runs in VISSIM with different random seeds provide variations in results. The necessary number of simulation runs (or sample size) can be established on the basis of confidence interval, sample Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 158 standard deviation and the allowed error range of a parameter being statistically analyzed. For example, Zong et al., (2002) suggested the following equation: (6.1) Where: n = required number of simulation runs a/2 = threshold value for significance level E = Allowed error range. However, the main purpose of simulation runs in this research is to generate random data for model testing rather than using these for statistical and analytical purposes. Therefore, the above suggested approach was not applied. If we refer to the mathematical models developed in previous chapters, we can see that they are all based on a statistical pattern recognition technique where the recognition of a neighbour is based on the similarities of its several features with that of a given focal point. In mathematical terms, each neighbour is a multi-dimensional point near the focal point. The dimensionality, the required sample size, and the recognition of neighbourhood are the difficulties to cope with for those who are applying statistical pattern recognition in practice. The difficulties are that at high dimensions, a point may be no longer the neighbour of a given focal point although it is so at low dimensions. This was referred to as “dimensionality curse” in the literature (Ernest, 1961, pp.94; Cleveland et al., 1988). For a better illustration, it should be interesting to read the following: “Regions o f high density may contain few samples, even fo r moderate sample sizes. For example, in the 10-dimensional unit multivariate normal distribution (Silverman, 1986), 99% o f the mass o f the distribution Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 159 is at points at a distance greater than 1.6, whereas in one dimension, 90% o f the distribution lies between ±1.6” (Webb Andrew, 2002, pp. 116). This means that if one wants to avoid non-recognition in a high dimensional space, the required sample size has to be increased. Table 6.4 presents suggestions by Silverman (1986) on required sample size based on dimensionality1. Table 6.4: Dimensionality and Required Sample size Dimensionality Required Sample size 1 4 2 19 3 67 4 223 5 768 6 2790 7 10700 8 43700 9 187000 10 842000 (Source: Silverman, 1986, pp. 94) For the developed model, the maximum dimensionality is three. Therefore, according to Silverman (1986), the required sample size should be more than or equal to 67 simulation runs. In this study, 100 and 70 simulation runs were considered for route 1 and route 95, respectively. The number of runs for each scenario is shown in Table 6.5. 1 Silverman (1986) suggested it for non-parametric density estimation in high dimensional space. Although our purpose is not the same, this reference would be valuable under the circumstance that information about required sample size for non-parametric regression in high dimensionality is difficult to find. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 160 Table 6.5: Scenario Design for Simulation Number of Scenario Situation Description simulation runs Route 95 Route 1 Simulate traffic networks with 1100 volumes 15% less than normal N/A 15 traffic 1000 Simulate network with normal 1200 40 20 traffic volumes Simulate network with volumes 1300 N/A 15 15% more than normal traffic Simulate traffic network with 2100 one-lane closed and normal N/A 15 2000 traffic volumes Simulate traffic network with 2200 slow-down zone with normal 15 20 traffic volumes Simulate traffic network with 3000 3000 busy stops and normal traffic 15 15 volumes. Total 70 100 Notes: N/A: Not applicable because the buses run on the dedicated bus lanes. As a consequence, ordinary traffic variations would have not influence on bus transit. Scenarios 1000 and 2000 were further divided into situations coded as 1100, 1200, 1300, 2100, and 2200. In the situations 1100 and 1300, daily traffic variations were simulated by increasing and decreasing normal traffic by 15 percent, respectively. Hourly traffic variations are based on the actual traffic counts and assumed to be the same for everyday. Situation 2100 is for one-lane closure due to road construction zone on the segment of Bank St. from Holmwood to Gladstone stops. This situation does not apply to bus route 95. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 Situation 2200 simulates slowdown zone. For bus route 1, the speed limit is as low as 10-12 km/h between Holmwood and Gladstone stops. For bus route 95, this zone is in between Hurdman and Campus stations where speed limit is 20-25 km/h. These slowdown zones were effective for 2 hours and removed after that time. Scenario 3000 simulates busy bus stops by increasing the number of boarding passengers by threefold. For bus route 1, these stops are Billings Bridge and Homlwood. For route 95, such stops are St. Laurent and Hudman. For each simulation run, 15 minutes (i.e., 900 seconds) were added to the simulation time as a warm-up time, the time to fill-up the empty traffic network with vehicles so that meaningless outputs are excluded. The state of the bus is set up to be updated at every 0.5 simulation second. This can keep the necessary accuracy while reducing simulation time and the magnitude of each output file2. The output file for each simulation was recorded by VISSIM in a table, namely OC TranspoVEHRECORD in the form of Microsoft Access database. Because the database is huge but only bus running times and passenger activities at stops are needed for the study, such information were filtered and re-tabulated. These tables are listed in Appendix D2. If we consider each VISSIM output after each simulation run as a record of a service day of the buses, then we have 100-day database for route 1 and 70 days of records for route 95. For these days, the situations mentioned in Table 6.5 may happen 2 Each simulation run by Intel Celeron Microprocessor 2.8 MHz, 1 GB RAM, and Window Home XP environment, is about 3600s for 8100s actual time with O.lsec-update set, and about 1400 seconds with 0.5 sec-update set. As a consequence, the magnitude of the file reduces from 32 MB to approximate 14-17 MB for each record, correspondingly. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 162 randomly, more or less frequently, due to the stochastic nature of traffic. In order to reflect these phenomena, the records were mixed up randomly. Also, by mixing the data, we could test the developed prediction model in term of its capability to recognize the similar bus running situations given a specified scenario. 6.4 Model Testing and Comparison Like several models discussed in literature review, the developed model in this study is devoted to real-time bus arrival and related predictions. In order to study the relative merits of the developed model, it is necessary to compare it with other models in term of prediction accuracy and reliability. However, this comparison does not mean that the developed model is more or less advanced than others. Actually, it is aimed at informing the readers about the strengths and weakness of the compared models in case these are to be implemented in the field. 6.4.1 Evaluation Criteria for Prediction Performance Three accuracy criteria shown in Table 6.6 are used to test the developed model as well as selected other predictors. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 Table 6.6: Accuracy Criteria for Model Testing Accuracy Criteria Equation Measurement MAPE provides unit-free MAPE measurement of the performance. Mean Absolute m a p e J ' y ^ - ^ l i o o It can be used to compare Percentage Error wtf *, V / predictors across different test sets. A IMRE MRE distinguishes the magnitude Maximum Relative Error MRE - m a x — ------1 xt of the error from a predictor RMSE I 1 * A RMSE measures the goodness-of- Root Mean Square Error U 5 > , - x ,)2 fit of the predictor. A Notes: x- actual data; X - predicted data; N- sample size 6.4.2 Reference Predictors As discussed earlier, for reference purposes, the developed model should be compared with some other predictors, given the same input data. Two predictors were selected. The first one can be considered as the simplest model, namely the naive model. The second reference model, which can be seen as one of the most advanced models in term of bus real-time arrival, was first developed by Reinhoudt e t al. (1997), and further applied by Farhan (2002), and Shalaby e t al. (2004). It is a Kalman filter-based model. The reason to choose the first reference model is that it is a simple model without any computational difficulties. The estimation of a bus operation (i.e., running time or passenger activity) between a pair of stops at current time is simply calculated by taking historical data of the bus itself and that of the same-day data just before the prediction time. The Kalman filter-based model aforementioned was chosen because it was successfully applied with both actual and simulated data for bus route 5 in downtown Toronto, Canada. Moreover, this model did outperform other models in a comparative Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 164 assessment (Shalaby et al., 2004), especially, the ANN-based model - which is a very powerful prediction tool. For the sake of illustration, brief presentations about these predictors are made below. The details can be seen in Appendix D3. Obviously, any prediction model that does not show consistent and more accurate forecast than the naive model would have little value. 6.4.2.1 The Naive Model This predictor is rather simple in concept. Naive predictions were calculated in a manner that take into account both current condition and historical bus running times or passenger activities as follows ' r(k- 1)^ r(k) = ■rhis(k) (6.1) rhis(k - V Where: r(k) and rhis(k) : running time or passenger activity of the bus for bus k at current time and one-day before. r(k-l) and rhiS(k-l): running time or passenger activity of the bus (k-1) at current time and one-day before. 6.4.2.2 The Kalman Filter-based Model The Kalman filter technique involves the state-space estimation. That is, the state of a subject in question at current time step is optimized (filtered) on the basis of the information about its historical states at previous discrete time steps and the latest Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 165 information. Reinhoudt et al. (1997), Farhan (2002), and Shalaby et al. (2004), used this technique in their research where the authors considered the new state being filtered is the bus running time between a pair of stops; the newest received information at the current time step is the running time of the same-day previous bus, which has just finished the run right before the bus in question. The historical data are the means and variances of running times recorded for the buses on several days before. According to Farhan (2002), the Kalman algorithm for bus running time prediction and passenger arrival rate is based on the system of four main equations as shown below. g(*+p= (6.2) e{k +1) = VARilocalj^ ]t+1 .g(k +1) ( 6 . 3 ) a{k +1) = 1 - g(k +1) ( 6 . 4 ) P{k +1) = a{k + Y).art(k) + g(k +1 ).artl (k +1) ( 6 . 5 ) Where g: filter-gain a: loop-gain e: filter error P: prediction art(k): actual running time or passenger arrival rate of previous bus at instant (k) artj(k+l) : actual running time or actual passenger arrival rate of the previous day at instant (k+1) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 166 VAR [dataiocai]k+i '■ average variance of the last three days’ data at time step k+1. If we take a closer look at this system, we can see that the determination of filter- gain (e.g., g(k+l)) plays the key role to update the prediction in Equations 6.4 and 6.5. To determine g(k+l), we need to know e(k) (Eq. 6.2), which in turn, requires g(k) to be defined (Eq.6.3). However, g(k) then needs e(k-l) to be estimated and so on. In other words, this is a recursive procedure where the bus running time being predicted at time step k requires an update, prediction, and correction (filter) procedure beginning from the very first trip running on that link (e.g. time step 1, or initial estimation). In his study, Farhan used 3 day-historical data for updating the variable VAR [dataiocaJ k+1. In this research, 3-day historical data were also used. An Excel code was developed to apply this algorithm for bus running time prediction. 6.4.3 Testing the Developed Model and Reference Predictors 6.4.3.1 Data issues The cross-validation technique is used in order to assess the accuracy of estimates. In this technique, the original sample data are mutually split into testing sample and learning sample. This technique is intended to avoid overly optimistic estimates when one uses the same data to construct the parameters for a prediction model and assess its performance. Although no parameters in our models are to be estimated (i.e., non- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 167 parametric model3), this technique is still applied in order to enhance the evaluations of model’s performance. There are various ways of performing data separation depending upon the magnitude of the original data. However, in a nutshell for the relatively small sample size that we have in hand, it is not practical to divide it in the way discussed above since it reduces the effective sample size. Instead, we consider the way based on a modified version of cross-validation technique, namely the leave-one-out cross validation (Hardle, 1990; Kwon et al., 2000). We treat one day as a test set and the remaining as historical data. By doing so, we can calculate the cross-validation evaluation criteria (e.g., MAPE, MRE, and RMSE) for the developed model and the reference predictors. In this study, 20 test sets, each for one time, were drawn randomly for the purpose of model testing. As the developed model combines the running time module and dwell time module, the performance of each module, as well as the whole model for bus arrival, was estimated separately and reported in the following sections. 6.4.3.2 Running Time Prediction Performance During the simulation time (i.e., 8100 seconds), each VISSIM record contained 9 trips4, 5 for bus route 1 and 35 trips for bus route 95. Consequently, we have 180 simulated trips for route 1 and 700 simulated trips for route 95, given 20 test sets for each route. Moreover, each trip belonging to route 1 has 3 running times (i.e., the time spent 3 Please refer to chapter 4, section 4.3 for discussions on parametric and non-parametric models. 4 Under this circumstance, each trip is defined when a bus departs at the first stop and arrives at the last stop of the studied segment For bus route 1, a trip starts when a simulated bus arrives at Billings Bridges stop and ends when the bus arrives at Rideau stop. For route 95, the two end stops are St.Laurent and Lebreton stops. 5 VISSIM uses the term “course” instead of “trip”. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 168 by each trip between a pair of understudied stops). Also, bus route 95 has 5 running times. In theory, if we apply cross-validation technique regarding every running time between all consecutive stops as well as every trip, we will have to cope with 540 and 3500 running time predictions for route 1 and route 95, respectively. These are actually large numbers and beyond the scope of this thesis research. In order to reduce the work load while we can maintain the generality in estimating the predictors (i.e., the developed model, the KF and the naive model), only specified trips and running times of each route are selected for model testing. For route 1, only the trips that start at 8:59 a.m. everyday at Billings Bridge station (i.e., trip 6) and only running times between Billings Bridge and Gladstone stops were considered. For bus route 95, only the trips that start routinely at 8:59 a.m. at St. Laurent station (i.e., trip 24) and only the running times between St.Laurent and Mackenzie stations were studied. By doing so, the number of predictions reduces to 20 for each route. To apply the running time module described in chapter 4, each 8:59 a.m. bus trip of route 1 was drawn from the 100-record database and considered as a test set. Running time of the 8:59 a.m. bus (i.e., trip 6) that is being predicted was defined by its pattern matrix containing three previous running times of 8:47 a.m. bus (trip 5), 8:37 a.m. bus (trip 4), and 8:27 a.m. bus (trip 3). The remaining data (i.e., 99 records) were treated as the historical pattern matrices. The differences between the actual running times and predicted running times in the test sets were calculated. The same procedure was also applied for the 8:59 a.m. bus trips at St.Laurent station of route 95. For the Kalman-filter based algorithm (KF), the record of 8:59 a.m. bus (trip 6) was drawn from the 100-record database and also considered as the test set. Based on the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 running time between Billings Bridges and Gladstone stops for trips 5, 4, 3, 2 ,1 and the historical running times experienced by 8:59 a.m. buses (trip 6) of 3 days ago, the running time was predicted and compared with the actual running time. For route 95, the same steps were applied for 8:59 a.m. bus trips running between St.Laurent and Mackenzie stations. In the case the Naive model, predictions were obtained from the use of Equation 6.1. The details for the use of data for testing the predictors are presented in Appendix D4. Figures 6.4, 6.5 and Table 6.7 present the results of cross-validation MAPE, MRE, and RMSE of the three predictors applied to route 95. Table 6.7: Cross-validation MAPE, MRE, and RMSE Cross-validation MAPE(%) MRE (%) RMSE (sec) criteria Developed Model 3.94 10.46 39.31 KF model 9.64 24.76 82.50 Naive model 15.49 90.53 181.73 l Developed Model ■ KF Model □ Naive Model 100.00 -90:53- 90.00 80.00 70.00 60.00 50.00 40.00 30.00 24.76 15.49 20.00 10.46 10.00 0.00 MAPE (%) MRE(%) Figure 6.4: Route 95- Running Time Prediction: MAPE and MRE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 170 ■ Developed Model ■ KF Model □ Naive Model 200.00 -484t73- 180.00 160.00 140.00 120.00 100.00 82.50 80.00 60.00 40.00 20.00 0.00 RMSE (sec) Figure 6.5: Route 95- Running Time Prediction: RMSE From Figures 6.4 and 6.5, we can see that the developed model is superior to the two reference predictors when they were all applied to the Transitway route (i.e., Route 95). The error measurements for the developed model are only a half and a quarter compared to those of the Kalman Filter and the naive predictor, respectively. This could be due to the fact that many types of impedance on bus running time were eliminated on the Transitway route so that the differences between situations are quite clear. Hence, the developed model recognized and classified the situations quite well. For route 1, a mixed-traffic bus route, due to many frictions that influence the bus running time, different bus running situations may be alike (i.e., they may have quite similar pattern matrices). Therefore, it makes it difficult for the developed model to cope with these cases when recognizing and classifying similar situations. This observation is supported by Figures 6 .6 , 6.7 and Table 6.8 when the developed model and the reference predictors were applied for route 1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 171 Table 6.8: Route 1- Running Time Prediction with Overall Situations: MAPE, MRE, and RMSE Cross-validation MAPE (%) MRE (%) RMSE (sec) criteria Developed Model 9.66 20.39 118.07 KF model 10.32 43.10 154.29 Naive Model 13.59 35.37 168.90 ■ Developed Model ■ KF model □ Naite Model 50.00 MAPE (%) MRE(%) Figure 6.6: Route 1- Running Time Prediction with Overall Situations: MAPE and MRE ■ Developed Model ■ KF model □ Naive Model RMSE (sec) Figure 6.7 Routel- Running Time Prediction with Overall Situations: RMSE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 172 Results shown in figures 6.6 and 6.7 suggest that the developed model for bus running time still outperforms the reference predictors since all indicators are favorable. However, the differences were not as remarkable as in the case of the Transitway route. In order to further explore the developed model, more comparisons were carried out. Fifteen running situations coded as 2200 (i.e., slowdown zone situation), each for one time, were extracted from database of route 1. It should be noted that these situations were mixed up randomly with other situations in order to reflect the variable nature of traffic as well as to test the developed model’s capability to recognize the similar situations in a mixed data. For each predictor, by comparing the prediction results of MAPE, MRE and RMSE regarding these slowdown zone situations with those of 20 overall situations mentioned previously, we can see that the accuracy of the KF predictor changed considerably. Its MAPE increases from 10.32% to 19.03%, the RMSE goes up from 154s to 291s (i.e., up to 88 %). In contrast, the prediction results of the developed model are still stable. The MAPE decreases from 9.66% to 8.76%. Furthermore, the MRE and RMSE are almost the same as they were. Figures 6.8 , 6.9, and 6.10 present these discrepancies. The comparisons point out that the developed model is more reliable than other predictors when dealing with abnormal traffic situations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 173 I MAPE (Overall) ■ MAPE(Slowdown zone) 25.00 23.62- 19.03 20.00 15.00 10.32 10.00 Developed Model Kalman Model Na'ft© model Figure 6.8: Route 1-The Differences of MAPE between the Overall and Slowdown zone Situations ■ MRE (Overall) ■ MRE(Slowdown-zone) 50.00 43.10 45.00 40.00 37.50 35.00 30.00 25.00 20.39 20.86 20.00 15.00 10.00 Developed Model KF model Naive model Figure 6.9: Route 1- The Differences of MRE between the Overall and Slowdown zone Situations ■ RMSE (Overall) ■ RMSE (Slowdown zone) Developed Model KF m odel Naive model Figure 6.10: Route 1- The Differences of RMSE between the Overall and Slowdown zone Situations Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The poorer prediction performances of the reference predictors are explainable. In order to predict a current bus running time, the KF and the naive predictors were based on the historical data of several days before that would have no running situations similar to the present one. For instance, to predict running time of a situation that has Dcode = 43 and Scode = 2200 in table D4.1 (see Appendix D4), the KF model was based on the running times of 3 previous days which have different situations with Dcode-Scode as 78-3000, 29-1300, and 87-1100, respectively. As a consequence, the increase of prediction errors of these models is understandable. Unlike the KF and the naive models, the developed model searched through the whole database, recognized the most similar situations, and gathered them for estimations. For example, with the same case above (i.e., the situation coded as 43-2200), the automatic neighbour searching procedure 6 searched through from 10 to 70 nearest neighbours around this case in order to find the optimal number of similar neighbours which provides the smallest cross-validation least-square prediction error (i.e. CR for short). It was found that 20 nearest neighbours offered smallest prediction error in this case, as shown in Figure 6.11. The figure was returned from the Matlab program coded for the developed model. Figure 6.12 depicts another example of such a neighbour searching for route 95, case 54-2200. 6 Please refer to chapter 4, section 4.4 for the details of this procedure. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 175 1 Ffle Ed* View Insert Toots D e s tto p W h d w Hefc> ** I Figure 6.11: Nearest Neighbour and Cross-Validation Least-square Prediction Error (CR) for Bus Running Time Prediction. Case 43-2200, Route 1. Insert Figure 6.12: Nearest Neighbour and Cross-Validation Least-square Error (CR) for Bus Running Time Prediction. Case 54-2200, Route 95. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 176 6.4.3.3 Boarding Passenger Prediction Performance As the number of boarders and de-boarders at the stops of route 1 are relatively small compared to that of route 95, only boarding passenger data of route 95 were used to test the predictors7. Also, for the sake of reducing computational work, only the number of boarding passengers at Mackenzie station of the buses that begin their trips at 8:59 a.m. (trip 24) at St. Laurent station was predicted. The boarding passenger prediction sub- module presented in chapter 5 (section 5.2.3) was applied for this purpose where the pattern matrix was defined by that of the previous bus (trip 20) served at Mackenzie station, the running time experienced by this bus, and the predicted running time of the bus itself (i.e., trip 24). The historical data of boarding passengers and running times of such buses were used as the historical pattern matrix. The KF model developed by Farhan (2000) was applied to estimate the passenger arrival rate at a stop. The number of boarders for trip 24 then was predicted by multiplying this rate and the predicted running time. To apply the naive predictor, Equation 6.1 was used. The details of each predictor are presented in Appendix D5. Figures 6.13 and 6.14 depict the cross-validation MAPE, MRE, and RMSE of the three predictors applied to forecast the number of boarders for 20 test sets. As shown in the figures, the developed model is the best, followed by the Kalman Filter and the naive predictor. 7 The average boarding passenger at stops of route 1 is only about 10-15 pas/ h /stop, or it equals to 3-4 pas/ bus/ stop. Therefore, even the difference between predicted and actual data of only 1 passenger/stop can cause a prediction error of up to 30% or more. This may lead to misjudgments on the predictors. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 177 I Developed Model ■ KF Model □ Naive Model 180.00 166.67 160.00 140.00 120.00 100.00 100.00 80.00 66.67 60.00 42.96 34.00 40.00 26.11 20.00 MAPE(%) MRE(%) Figure 6.13: Route 95 - Boarding Passenger Prediction: MAPE and MRE ■ Developed Model ■ KF Model □ Naive Model passenger RMSE Figure 6.14: Route 95 - Boarding Passenger Prediction: RMSE Figure 6.15 shows an example of a neighbour searching result for a boarding passenger prediction. It should be noted that while the KF predictor needs the controversial assumption of a uniform passenger arrival rate, there is no such assumption for the developed model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 178 Figure 1 - jiO l f x l He Cdft T u o b wll lHODW h iliim » ncp »-j— Dfi? B S ] > | O ' ® ;« me s'D ■ 1111 . Figure 6.15: Nearest Neighbour and Cross-Validation Least-square Error (CR) for Boarding Passenger Prediction. Case 24-1200, Route 95. 6.4.3.4 Alighting Passenger Prediction Performance The number of alighting passengers of the buses at Mackenzie station that start at 8:59 a.m. (trip 24) from St.Laurent station was used for testing the predictors. In doing so, the alighting passenger prediction sub-module in chapter 5 was used. Following the sub- module, alighting passengers of the two previous buses (i.e., trips 20 and 19) that served at Mackenzie stop were used as the elements of the current pattern matrix defining the de-boarders of trip 24 that are being predicted. The historical data of alighting passengers of these buses was used as the historical pattern matrix for neighbour searching and prediction purposes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 179 The Kalman Filter algorithm developed by Farhan et al., (2002) was not originally developed by the above authors for alighting passenger prediction. In this study, this method was modified with minor changes so that it can be used for alighting passenger prediction. For the nai've model, the predictions were based on Equation 6.1. The details of each predictor are presented in Appendix D 6 . Figures 6.16 and 6.17 depict the MAPE, MRE, and RMSE of the 3 predictors applied to forecast the number of de-boarders. i Developed Model ■ KF Model □ NaiVe Model 90.00 80.00 70.00 59.09 60.00 50.00 40.00 30.00 24.65 20.00 MAPE(%) MRE(%) Figure 6.16: Route 95- Alighting Passenger Prediction: MAPE and MRE B Developed Model ■ KF Model □ Naive Model Figure 6.17: Route 95- Alighting Passenger Prediction: RMSE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 180 Figure 1 I - J lj n X ; Hie Edt Vfew Insert Tools Desktop Window Hefc U D H i k | ** f? ® * I OB] Figure 6.18: Nearest Neighbour and Cross-Validation Least-square Error (CR) for Alighting Passenger Prediction. Case 20-1200, Route 95. The developed model outperformed the comparative predictors in term of alighting passenger prediction as shown in Figures 6.16 and 6.17. Figure 6.18 shows the result of an example of a neighbour searching. 6.43.5 Prediction Performance with Actual Data Running time data and the information of the number of boarding and alighting passengers retrieved from the APC and AVL systems embedded on the buses of route 1 and route 95 were used to test the predictors. Figures from 6.19 to 6.26 present the MAPE, MPE, and RMSE of the 3 predictors applied to the selected segments and stops of route 95 and route 1 with different test sets. Please see Appendix D7 for details. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 I Developed Model ■ KF Model □ Naive Model 120 104.24 100 80 SS 60 40 26.72 22.45 20 14.12 0 MAPE {%) MRE(%) Figure 6.19: Route 95 - Running Time Prediction with Actual Data: MAPE and MRE I Developed Model ■ KF Model □ Naive Model Second 200 180 160 140 120 98.52 100 80 60 45.03 40 20 0 RMSE Figure 6.20: Route 95 -Running Time Prediction with Actual Data: RMSE ■ Developed Model ■ KF Model □ Naive Model 45.00 39.10 40.00 35.00 32.35 30.00 25.00 20.00 15.78 10.00 MAPE(%) MRE(%) Figure 6.21: Route 1-Running time Prediction with Actual Data: MAPE and MRE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 182 l Developed Model ■ KF Model □ Naive Model second 200.00 176.44 180.00 160.00 140.00 120.00 99.91 100.00 80.00 63.20 60.00 40.00 20.00 RMSE Figure 6.22: Route 1-Running Time Prediction with Actual Data: RMSE I Developed Model ■ KF.Model □ Naive Model 300.00 250.00 240.00 200.00 150.00 100.00 83.33 69.14 46.15 50.00 37.39 20.18 0.00 MAPE(%) MRE(%) Figure 6.23: Route 95 - Boarding Passenger Prediction with Actual Data: MAPE and MRE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 183 a Developed Model ■ KF.Model □ Naive Model Figure 6.24: Route 95 - Boarding Passenger Prediction with Actual Data: RMSE I Developed Model ■ KF. Model □ Naive Model 70.00 66.67 50.00 38.46 40.00 30.00 23.08 20.00 12.39 10.00 MAPE(%)MRE(%) Figure 6.25: Route 95 - Alighting Passenger Prediction with Actual Data: MAPE and MRE a Developed Model a KF. Model a Naive Model RMSE Figure 6.26: Route 95 - Alighting Passenger Prediction with Actual Data: RMSE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 184 The figures reveal that the developed model has the best prediction performance compared to the reference predictors. For running time prediction, the average MAPE is only about 6% for both routes. The MRE varies between 10% and 11%, and the RMSE is in the range from 45 to 65 seconds. For boarding passenger predictions, the developed model’s MAPE and MRE are as low as 20% and 46%, respectively. For alighting passenger predictions, its MAPE and MRE are 8 % and 23%. All indicators reflect a favorable prediction model. 6.4.3.6 Tukey Test for Performance Comparison The prediction performances of the three predictors were evaluated with different test sets. Although the charts presented earlier visually depict that the developed model outperformed the reference models, the comparisons based on the sound statistical analysis are the basis for reliable conclusions. In this part, Tukey’s test was used to compare the MAPE values of the predictors. The details of Tukey procedure are presented in Appendix D 8 . Tables 6.9 and 6.10 present the results. The results of the Tukey’s procedure prove that the developed model has smallest MAPE compared to that of Kalman Filter and the naive models in most of the test sets at 95% confidence level. Therefore, the developed model can be considered as the preferred prediction model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 185 Table 6.9: Studentized Range of the Test Sets Test sets Q 0.05. 337 MSE w Route 95: Simulated data of running times 3.42 15.1 2.97 Route 1: Simulated data of running times 3.42 8.72 2.25 Route 1: Simulated data of running times for 3.43* 7.78 2.47 slowdown-zone scenario Route 95: Simulated data of boarding passengers 3.42 79.74 6.82 Route 95: Simulated data of alighting passengers 3.42 23.67 3.72 Route 1: Actual data of running times 3.43* 6.55 2.26 Route 95: Actual data of running times 3.43* 13.79 3.29 Route 95: Actual data of boarding passengers 3.43* 114.46 9.47 Route 95: Actual data of alighting passengers 3.43* 16.91 3.64 Notes: MSE: Mean Square Error within groups; q: Studentized range distribution value (Netter, 1985); w: range * Resulted from q aes, 3,42 Table 6.10: Results of Tukey’s Procedure at Significance Level = 0.05 MAPE (%) TETS SETS Developed KF.Model Naive Model Model Route 95: Simulated data of running times 3.94 9.64 15.49 Pairwise Comparison 5.70 5.85 Route 1: Simulated data of running times 9.66 10.32 13.59 Pairwise Comparison I 0.66 3.27 Routel: Simulated data of running times for 8.76 19.03 23.62 slowdown zone scenario Pairwise Comparison 6.94 3.28 Route 95: Simulated data of boarding passengers 26.11 34.00 42.96 Pairwise Comparison 7.89 8.96 Route 95: Simulated data of alighting passengers 5.73 24.65 14.59 Pairwise Comparison | 8.86 10.06 Route 95: Actual data of running times 6.6 10.26 26.72 Pairwise Comparison 3.66 16.46 Route 1: Actual data of running times 6.02 8.38 15.78 Pairwise Comparison 2.36 7.40 Route 95: Actual data of boarding passengers 20.18 37.39 69.14 Pairwise Comparison 17.21 31.75 Route95: Actual data of alighting passengers 8.28 12.37 20 Pairwise Comparison 4.09 7.63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 186 6.4.3.7 Bus Arrival Time Prediction Performance Each component of the developed model was statistically tested and the performance results are encouraging. Therefore, the author is confident that the integrated model will outperform the reference predictors when applied to bus arrival time predictions. In order to estimate performance of the integrated model for bus arrival time predictions, 15 full trips of route 1 that start at Billings Bridge station at 8:59 a.m. were selected randomly. For each trip, running times, dwell times, the number of boarding and alighting passengers, departure times, and arrival times of the buses at four selected stops (i.e., Billings Bridge, Holmwood, Gladstone, and Rideau stops) were predicted. To predict bus arrival times at the stops, besides the use of the developed modules and sub-modules for predicting and updating bus running times on the links and dwell times at the stops, the updating predictions procedure presented in chapter 3 was applied in order to update arrival and departure time information of the buses at stops. The details of these predictions can be seen in Appendix D9. Tables 6.11, 6.12 and 6.13 display the performance of the three predictors in term of bus arrival time at Holmwood, Gladstone and Rideau stops. Table 6.11. Holmwood Stop: Bus Arrival Time Prediction Performance of the 3 Predictors Cross-validation MAPE (%) MRE (%) RMSE (sec) criteria Developed Model 8.60 20.14 62.31 KF model 12.43 43.10 81.77 Naive Model 12.63 57.42 118.13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 187 Table 6.12. Gladstone Stop: Bus Arrival Time Prediction Performance of the 3 Predictors Cross- MAPE (%) MRE (%) RMSE(sec) validation Criteria 1st update T d update 1st update 2nd update Ist update 2nd update Developed 5.58 2.93 16.82 7.95 85.18 40.69 Model KF. model 8.93 3.85 17.24 8.79 97.86 50.95 Naive 10.53 7.46 42.64 23.41 160.53 100.90 Model Table 6.13. Rideau Stop: Bus Arrival Time Prediction Performance of the 3 Predictors Cross- MAPE (%) MRE (%) RMSE (sec) validation Criteria Is' 2nd Last 1st 2nd Last 1st 2nd Last update update update update update update update update update Developed 5.25 3.32 2.58 12.75 11.87 4.17 97.16 70.43 47.19 Model KF. 6.27 4.52 3.99 15.04 9.41 8.68 120.80 68.61 60.25 model Naive 10.22 6.17 6.87 27.11 12.33 12.55 196.63 123.64 127.33 Model The results presented in the tables suggest that the developed model is the most preferable model, followed by the KF and the naive model. Also, the predicted arrival times at the stops become more accurate when the most updated bus information became available. For example, at Rideau stop, the average of MAPE of the 15 full-trips decreased from 5.25% to 3.32% and 2.58% corresponding to the first, the second and the last update. Table 6.14 shows a part of real-time prediction results for bus arrival times at the stops. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 188 Table 6.14: Numerical Examples of Real-time Bus Arrival Prediction Using Developed Model for 3 full trips (seconds) Holmwood Gladstone Rideau Updated 544,37 970.24 1583 09 1st 953.34 1581.21 1590.66 Actual Arrival Time at Stop 1591 Holmwood Gladstone Rideau Updated 579.64 1274.42 1891.46 1st 1336.06 2033.75 2017.58 Actual Arrival 2001 Holmwood Gladstone Rideau Updated 548.47 1191.79 1793 04 1st 1193.45 1783.68 1870.02 1848 6.5 Summary Different test sets were used to test different constituent modules as well as the integrated developed model in term of prediction capability. In order to compare the developed model with other predictors, the Kalman filter-based model selected from the literature and the naive model were also used to predict bus running times and passenger activities. It was found that all modules of the developed model were statistically superior to the reference predictors, given the same input data. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 189 When applied for obtaining running time predictions, the running time prediction module worked satisfactorily, given its MAPE of only nearly 5% and 7% on average when using both actual and simulated data for route 95 and route 1, respectively. The real-time boarding passenger and alighting passenger sub-modules also worked quite well and all outperformed the reference predictors. The MAPE for boarding passenger prediction was from 20% to 26% and from 6% to 8 % for alighting passenger estimation. It should be recalled that while other prediction models mentioned in the literature review need a controversial assumption on passenger arrival rate, the developed model does not require this assumption. While the reference models showed instability when dealing with unusual event occurring in bus service activity, the developed model showed only small changes in its prediction performance. That is, it worked consistently. For real-time bus arrival time prediction, the integrated model produced good results as can be appreciated with average MAPE as low as 3%. The updating method helps to improve the accuracy of the model when new information becomes available and fed into the model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 7 REAL-TIME PREDICTION INTERVAL, ON-LINE ADHERANCE EVALUATION AND BUS BUNCHING DETECTION 7.1 Introduction In this chapter, methods to define real-time prediction interval, bus on-time performance, and bus bunching detection are reported. In addition, further applications of the developed model are discussed. Finally, the spatial aspect and the level of detail for database are mentioned in order to facilitate model efficiency. 7.2 Real-time Prediction Interval Prediction is a challenge and will never be a fixed value. Therefore, a prediction interval is important under any circumstances. In this study, prediction interval for bus running time carries the evidences of non-parametric regression and thus, it is complicated and time consuming. Several methods have been developed to determine the prediction interval for non-parametric regression family. These methods can be seen in the books of Hardle (1990), Fan et al., (1996) or Loader et al., (1999). However, only the methods that are theoretically sound and can be applied easily are of interest in this research. In this 190 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 191 section, an automatic computer-intensive procedure was applied to define prediction interval dynamically with each prediction. According to Fan et al., (1996) the prediction intervals for pointwise locally weighted regression can be expressed as follows. A A fA 1 1/2 mj(x0) ~ f.bj,p(x0) ± z ^ a/2 .j\< Vj,P(x0) j> (7.1) Where: A ntj (x0) and j have been explained in chapter 4 z/_o/2 denotes the (l-«r/2)th quantile of the standard Gaussian distribution A A bj,p(x0) , Vj,P(x0) the estimated conditional bias and variance The estimates of bias and variance are obtained by calculating expectation and variance from equation defining P as follows (Please see Equation 4.7, chapter 4) E (^(x 0) jX l,X 2,...,Xn) = (X W X /,X Wm = p + (X? WX/ 1 X W r (7.2) Var(i3{x0) | X x,X 2 ,...,Xn) =(XtW X/' (XlIX) (X WXJ1 (7.3) Where: m-(m(Xi),m(X2), ...,m(X^f, ft=(m(xo), (xo)/plf and r =m -X fi, the residual of the local polynomial, and II is square covariance diagonal matrix with 77/,, = K 2(^ ‘ — ) The bias in equation 7.2 and the variance in equation 7.3 cannot be directly usable because of unknown quantities r and II. Therefore, they need to be approximated A A through bj,p(x0) , Vj,p(x0) which are respectively the estimated conditional bias and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 192 A variance. These are the decompositions of the estimator of mean squared error ( MSE ) of A A P j = mj(xQ)/ j\ as shown below M SE jA x0 ,h) = b2 jA x 0) + V jA x 0) (7-4) According to Loader et al. (1999), the estimated bias and variance can be defined by the following equations. A A ^ Vi Ax,) ~ er (x0) (7.5) Where: A 2 1 " A Y _ r o- (*o) = ------r ----- — YW -YifJCi-^—^) (7.6) 0 tr(W)-tr((X,WX)-iX ‘W 2X ) t i h bj,P (x0) = (X'wxy]X 'W r (7.7) Where: r is a row vector which replaces the residual vector in equation 7.2 in a higher order of polynomial. While the estimator of variance can be approximated easily, it was found that the estimator of bias shown in equation 7.7 is unrealizable as it is based on a larger order of regression function with different pilot bandwidths (Fan et al., 1996). Therefore, it is an expensive and complicated computation. In chapter 4, we applied leave-one-out method to define an optimal bandwidth, A given a focal point. Based on that bandwidth, the estimator of mean square error MSE was defined as a by-product of the bandwidth computation. As thus, we can estimate the estimator of bias by using the system of equations 7.4, 7.5 and 7.6. Once the estimators Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 193 of bias and variance have been determined, the prediction interval can be defined. The steps to determine a prediction interval are shown below: Step 1: Determine the optimal bandwidth by leave-one-out method presented in chapter 4. Step 2: Based on the optimal bandwidth and the recognition of the neighbours around the given focal, determine the local regression function. Step 3: Based on the local regression function, calculate mean squared error A {MSE) of the local neighbours in the domain of the optimal bandwidth. Step 4: Determine the estimator of variance based on equations 7.6. A Step 5: Determine the estimator of bias based on MSE and the estimator of variance. Step 6 : Estimate prediction interval based on equation 7.1 with z statistic replaced by t statistic due to the approximation of variance with the degree of freedom shown below (Fan et al., 1996). (7.8) The overall procedure was coded in Matlab. Please see Appendix C. Table 7.1 presents pointwise prediction intervals for running time test sets of route 1 with two levels of significance at 0.05 and 0.01. As seen in the table, the predicted values were not the center of corresponding prediction intervals due to bias. The predicted values tend to be closer to the upper bounds of their prediction intervals. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 194 Table 7.1: Real-time Prediction Interval for Bus Running Time of Route 1 ______(Time in second)______Predicted a=0.05 a=0.01 RCODE* Running time lower bound upper bound lower bound upper bound 1 946 764 1003 707 1060 7 962 735 1025 656 1095 12 923 740 984 682 1041 17 925 672 1014 591 1094 18 949 755 1015 695 1075 23 935 699 1007 629 1073 28 859 606 950 525 1025 33 912 718 975 659 1033 37 951 751 1015 690 1076 40 828 638 895 578 955 46 896 655 975 579 1050 50 909 742 967 688 1020 57 899 596 1006 499 1103 61 904 733 959 680 1012 68 954 724 1033 651 1070 * RCODE: a unique code set by the author for the purpose of computer programming With a real-time prediction interval, a bus transit planner can capture the variation domain of the predicted bus running time, given a certain confidence level. Therefore, the provision of real-time prediction interval enables other applications of the developed model in the bus transit field. The following sections present the applications of this method for defining probabilities of the bus being on-time or in pair form. 7.3 On-line Schedule Adherence Evaluation 7.3.1 Methodology As discussed in previous chapters, bus service providers and passengers need to know ahead of time whether the bus will be on-schedule or not. The model developed in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 195 this thesis research allows transit providers to know about bus arrival time as well as the prediction interval in real-time at any stop. However, the methodology to determine the probability of bus being on-time has yet to be developed. Data for every 3 monthly periods that were retrieved from the AVL-APC systems in the OC Transpo were used to estimate bus on-time performance. Comparisons were made of the retrieved bus arrival times at bus stops with the scheduled times and then the differences were statistically analyzed. Such a difference is called the off-line adherence performance (OAF) of the bus. The OC Transpo as well as other bus transit agencies throughout North America consider that the bus is on-time if its OAP falls between 1 minute of earliness and 3 minutes of lateness (Stramathan et al., 1999; Shara, 2002). We call these values as the on-time limits and the time between the limits is on-time standard range. To forecast the probability of the bus being on time in real-time, it is easy to apply the above on-time standard range, given a real-time bus arrival prediction. Now we call the difference between a scheduled arrival time and a real-time arrival time as Real-time Adherence Performance (RAP). Is the bus on-time given a predicted RAP? A simple answer is that if this RAP is within the on-time standard range, then the bus can be considered as on-time. However, in order to provide bus dispatchers more information about on-line arrival performance of the bus, further steps should be taken as follows 1. At each stop, the history of bus arrival times is collected and compared with scheduled arrival time. 2. Examine the density distribution of OAP by assuming that it follows one of the following distributions: the Gaussian or Lognormal distributions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 196 3. Find the best fit distribution using x 2 (Chi-square) test or Kolmogorov-Smimov test (K-S test). 4. Determine the RAP given a predicted bus arrival time and its prediction interval 5. Calculate the probability of the bus being on-time and other possibilities, given the best-fit distribution, the RAP, and the prediction interval. 7.3.2 A Numerical Example For the sake of illustration, we select arrival times recorded at Gladstone stop of the buses that are scheduled to depart routinely at 8:59 a.m. (i.e., at the 1st second) at Billings Bridge station and to arrive at Gladstone stop at 9:10 a.m. (i.e., at the 660th second). The data were retrieved from the APC systems for one year period from 2004 to 2005. 1. The differences between historical arrival times and scheduled arrival times {OAP) were calculated and arranged in the increasing order. 2. To fit the OAP distribution, the Gaussian and Lognormal probability distributions were applied. Equations 7.8 and 7.9 present these distributions, respectively. Lognormal Distribution (7.8) (x - miri m = 0 if x < min Gaussian distribution (7.9) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 197 Where: min = minimum x A h = interval length; A h = (maximum x - minimum x)/total classes g = Location parameter of a lognormal distribution 0 = shape parameter a = standard deviation ft = mean 3. A null hypothesis (H0) that the Gaussian or the Lognormal probability functions provide good fits for the distribution o f OAP was made and tested. For the chi- square test, let’s call x 1 as the chi-square value estimated from the goodness-of-fit procedure. If ^ 2e« is greater than^2cnw at a= 0.05, then H„ is rejected, meaning a poor fit. Otherwise, if %2eti is smaller than % 2 critical, Ho cannot be rejected (DO NOT REJECT), or a good fit is recognized. For the K-S test, the K-S estimated will be compared with K-S critical at a= 0.05. If the K-S estimated is smaller than K-S critical, then Ho cannot be rejected (DO NOT REJECT). 4. For each type of distribution, the theoretical frequencies were calculated and compared to the observed frequencies of OAP. These discrepancies were then used to estimate % 2 and Kolmogorov- Simimov test. Table 7.2 presents the results of the fitting procedures. These are the products of the StaFit 2.0 software. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 198 Table 7.2: Goodness-of-fit Test Summary (a=0.05) data points 29 estimates moment estimates accuracy of fit 3.e-004 level of significance 5.e-002 Summary distribution Chi Squared Kolmogorov - Smirnov Lognormal 0.855 (3) 0.149 Normal 0.746 (3) 0.176 Detail Lognormal Distribution minimum = 105. [fixed] mu = 4.63621 sigma = 0.615371 Chi Squared total classes 15 interval type equal length net bins 4 chi**2 0.855 degrees of freedom 3 alpha 5.e-002 chi* *2(3,5.e-002) 7.81 p-value 0.836 result DO NOT REJECT Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 199 Kolmogorov-Smirnov data points 29 K-S stat 0.149 alpha 5.e-002 K-S stat(29,5.e-002) 0.246 p-value 0.495 result DO NOT REJECT Gaussian distribution mean = 229.655 sigma = 83.107 Chi Squared total classes 15 interval type equal length net bins 4 chi* *2 0.746 degrees of freedom 3 alpha 5.e-002 chi**2(3,5.e-002) 7.81 p-value 0.862 result DO NOT REJECT Kolmogorov-Smirnov data points 29 K-S stat 0.176 alpha 5.e-002 K-S stat(29,5.e-002) 0.246 p-value 0.296 result DO NOT REJECT Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 200 5. The results show that both distributions were successfully in fitting the actual data at a=0.05 (see Table 7.2 and Fig.7.1) but the Gaussian distribution gave a better fit. Therefore, the best-fit probability function is: 29 , (x -229.655) \ /(* ) = ex p (------J—) (7.10) 83.107 4 l7 t 2(83.107) Fitted Density for OAP at Gladstone stop 100. 20 0. hpitBLo^wrmaliliQniHl Figure 7.1: Fitted Density for OAP at Gladstone Stop- Route 1 Assume that a real-time prediction of bus arrival time, together with its prediction interval, has just been made as 946th second at Gladstone stop (e.g., the case with RCODE=l, a= 0.05, Table 7.1). The scheduled arrival time for this bus is at 660th second. We calculate the differences between the lower prediction bound (764th second), the predicted value (946th second), and the upper prediction bound (1003rd second), with the scheduled arrival time. These are Xi=104s, X 2 = 286s, and X 3=343s, respectively. As Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 201 discussed earlier, the bus is considered to be on time if the predicted arrival time compared to schedule times is 60 seconds (1 minute) of earliness or 180 seconds (3 minutes) of lateness. We denote the on-time limits as Oi (lower limit) and O2 (upper limit). By applying Equation 7.10 with the values of X and O, we can have the probabilities that the bus will be on-time or not. Figure 7.2 depicts the locations of those points. 0.20 , - „ Frequency Part o f on-time standard range Real-time Prediction Interval 600. OAP (second) Figure 7.2: On-time Limits and Prediction Interval In Figure 7.2, the lower limit Oj is not presented since the lowest value of OAP is 100 seconds of lateness. Therefore, there is no chance that the bus is on-time at the stop during the time range of 600th ~76(fh second. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 202 Because the predicted RAP (i.e., X2=286 seconds) is larger than the on-time upper limit (02=180 seconds) the bus is very likely not on-time. However, as the prediction lower bound ( Xi=104 seconds) is smaller than the upper limit, there is still a chance the bus is on-time. The probability that the bus is on-time and its RAP falls between the range X j0 2 (i.e., 104-180 seconds) is exactly the area bounded by the distribution curve (Equation 7.10) and part of the horizontal axis X \0 2 as found below: P(104 < RAP < 180) = 83.107a/2 tt In other words, the probability that the bus is on-time is 21%, with the bus arrival time is in between 764th ~ 840th second. Similarly, the probability that the bus is behind the schedule and the lateness is within 180 to 343 seconds (0 2X3) is as follows 343 P{\ 80 < RAP < 343) = J 180',83 .10772* If we convert the RAP into the arrival time, the probability that the bus is late is 62% with the bus arrival time in between 841st ~1003rd. In general term, the probability that the bus is not on-time is 79%. That is high enough to conclude that the bus is not on-time. 7.3.3 Discussion 1. Unlike the simple answer that the bus is certainly not on-time (e.g., probability that the bus is not on-time is equal to 1), if a predicted bus arrival time is out of the on- time standard range, the above example showed that there is still a chance the bus is on- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 203 schedule. This happens when an overlap between prediction interval and standard range exists. If the probability that the bus is not on-time is higher than that of the bus is on- time, there is potential that the bus is not-on time and vice versa. 2. If there is no overlap and the prediction upper bound X 3 is smaller than the on- time lower limit (Oj), then the probability that the bus is not on-time and early is 1. Similarly, if prediction lower bound X\ is larger than the on-time upper limit (02), then the probability that the bus is not on-time and is late is 1. 3. By combining the method to predict real-time prediction interval in previous section and the methodology developed in this section, bus dispatchers can quantitatively analyze all possibilities of bus adherence to the schedule. This would help them to make better decisions in managing their bus fleet. 7.4 Real-time Bus Bunching Detection Method 7.4.1 Methodology Once bus arrival times at every stop can be predicted in real time, the tendency of buses to form in pairs (or bus bunching) can be detected. Knowledge in advance of a bus bunching will help bus dispatchers to make a proactive solution to prevent it. The decision should be based on the probability that bunching may take place and the experience of the bus dispatchers. In this section, a method to provide that probability is developed. Given the OAP distributions of bus k and k+1 at a specified stop, the predicted bus arrival times of bus k and bus k+1 , and their prediction intervals, the probability of a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 204 bus bunching at the stop is positive if the upper prediction bound (AR.3,1) of bus k is larger than the lower prediction bound (ARj, *+/) of bus k+1 as shown in Figure 7.3. Otherwise, this probability is equal to zero as illustrated in Figure 7.4. Frequency fk(x) On-time Performance ARi,krSc k AR 2fkrSc * ARjr t+r Sc *+; AR 2>k+i -Sc k+i AR 3, k+i - Sc k+i Figure 7.3: Tendency of Bus Bunching, P>0 F requency ft+i(x) On-time Performance Figure 7.4: Tendency of bus Bunching, P=0 In Figure 7.3, the probability of bus bunching can be derived as follows: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 205 Where: A R ij and A R ij (j=k, k+1 ) = the lower and the upper prediction bounds of a prediction interval of bus j Sck and Sck+i = scheduled arrival times for bus k and k + 1 fk(x) and fk+i (x) = probability functions of OAP of bus k and k+1 at the stop Obviously, the larger the Pb, the closer the bus dispatchers should pay attention to this phenomenon. The developed method for bus bunching detection is summarized as follows 1. Determine a pair of buses needed to be examined for bunching at a certain stop (e.g., bus k and bus k+1 ). 2. Use the developed model to predict bus arrival times and the corresponding prediction intervals for each bus. 3. Determine the goodness-of-fit functions of OAP for bus k and k+1 at the stop (e.g., fk(oA P ) andfk+j(OAP)). 4. Compare the upper bound prediction of the preceding bus (AR3, k) with the lower prediction bound of the following bus (AR/ k+i). If ARhk+i > AR3, k then Pb~ 0 or there is no chance the two buses can form a pair. If A R j, k+i < AR 3, k then Pb is found from Equation 7.11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 206 7.4.2 A Numerical Example To illustrate the methodology described in the previous section, a pair of 8:56 a.m. and 8:59 a.m. buses departing from St.Laurent station of route 95 was selected. The scheduled arrival times of these buses at Mackenzie stop are 9:05 a.m. and 9:08 a.m., respectively. Historical data of bus arrival times recorded by the APC systems were used to examine the OAP distributions of the two buses at Mackenzie stop. The real-time arrival times and the prediction intervals of these buses were predicted by using the developed model. We have to predict if the buses are forming into a pair or not. The parameters of the two buses are presented in Table 7.3. Table 7.3: Parameters of the Two Consecutive Buses Parameter 8:56 a.m. bus trip i 8:59 a.m. bus trip 36 [ln( j - 30) - 4.2439 ]2 50.5 [ln(x + 135) - 5.1567 ]2 M ) 0.7598 Off-line Fitted Density for OAP at Mackenzie stop Fitted Density for OAP at M ackenzie stop Adherence fc35 0.25 Performance function at Mackenzie e.13 station i f c i j L i L . tooo a s -too. iso. 200. 2» . 30&. -290. -W0. 0 J « W0. 200. 300. 000. kautVSiiM Input Vatuc* |* k*Mi! ■ Lognornm ■ Bonnet | |« Input ■ Lognornnl ■ aocnHl | Scheduled Sck+i= 720fh second Arrival Time Sck= 54Cfh second 60 ~ +180 seconds ( equivalent to 60 ~ + 180 seconds ( equivalent On-time limits arrival time in between 48(fh ~ 720th to arrival time in between 660th second) ~ 900th second) Real-time Lower bound ARj,k= 611th sec Lower bound AR / * + / = 724th sec Prediction of Predicted arrival time = 748th sec Predicted arrival time = 879th sec bus arrival time Upper bound ARs ^ 794th sec Upper bound ARstk+i= 898th sec Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 207 As shown in Table 7.3, AR^ * is larger than ARty *+/ so there is a chance that the two buses will form a pair at Mackenzie stop. The probability of this incidence is calculated as follows ARiJ.-Sck AR3J[-Sck+t 794-540 794-720 Pb = J fk(x)dx. j f k+l(x)dx =J f k(x)dx. j f k+l(x)dx = ARiM \ ~Sct 724-540 724-720 Pb = 0.067*0.235=0.0154 or 1.54 %. The probability is too small to be concerned about bus bunching at Mackenzie stop. 7.5 Further Possible Applications by Using the Developed Model This section focuses on further possible applications of the developed model in the field of bus transit control. Instead of detailing the control methods currently used in transit agencies since it may lead to another field of bus transit and therefore divert the focus of the thesis out of its targets, this section aims at listing some real-time control models whose critical bases are the availability of real-time bus arrival and dwell time information at bus stops. The bus control strategies are moving towards real-time responses to the sporadic service problems (Strathman et al., 2000). In the real-time bus control strategies, vehicle holding, short-turning, stop skipping, and speed modification are the tools to maintain the headway and service reliability given the knowledge of bus service problem in advance such as lateness, earliness or bunching. Having assumed that real-time bus running times and dwell times are available, many researchers did develop models for real-time bus Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 208 control strategies for different purposes. The readers should refer to the work of Strathma et al., (2000) for automatic bus dispatching model, or the work of Ding et al., (2002); and Fu et al.,(2003, 2004) for optimizing headway variance. Those who are interested in speed modification can read the work of Shalaby et al., (2004). For service reliability or passenger utility, the works of Tumquist (1982) and Rabi et al., (2000) are suggested. 7.6 Real-time Bus Arrival Information Broadcasting When arriving at a bus stop, passengers need arrival information of the next bus (buses) that is supposed to be broadcasted. Because broadcasted time changes as real time arrivals are updated, passengers may get one or more than one bus arrival announcements. In the developed model, the predicted bus arrival time of the current bus is based on the running times of previous buses. Therefore, if the headway is long, the prediction may be inaccurate. The author suggests that if the headway between buses is longer than 30 minutes, the real-time prediction should not be made and the announced arrival should be “no real-time information”. For the incoming stop (i.e., the stop closest to the bus), besides the real-time bus arrival time being announced, bus transit agencies should consider broadcasting the information about bus on-time possibility. The bus should be announced as “on-time” when the probability that the bus is on-time is larger than it is not so1. Otherwise, it should be broadcasted as “not on-time”. For instance, in the numerical example in section 7.3.2, the probability that the bus is on-time is 21%, while the probability that it is not so 1 The method to estimate this probability was developed in section 7.3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 209 is 79%. Hence, the announcement at Gladstone stop should be as “Next bus is not on- time and will arrive in 4 minutes2”. This would help passengers to know clearly that the expected bus is coming or it has already passed the stop. 7.7 Level of Spatial and Temporal Detail of Data for the Developed Model A sound and well organized database is needed in order to facilitate the applications of the developed model. This section discusses a key aspect of a database for the model, the requirement of data capture and database design. Table 7.4: Level of Spatial and Temporal Details for Data Capture Event- Between-stop Level Description Independent Event Records Performance records Data AVL data Infrequency A without real (60-120 sec) time Tracking AVL data with Infrequency each timepoint B real-time (60-120 sec) Tracking APC and event each stop C recorder Event recorder Each stop and Recorded D with between in between stop events and stop summaries summaries Event All events, full E recorder/trip Very frequent all types speed profiles recorder Furth et al., (2003) concluded that there are five levels of spatial and temporal detail, as illustrated in Table 7.4. Referring to the table and the data contents that the 2 Number 4 is only an example. In fact, the announced time is the difference between the predicted arrival time and the current clock- time when the arrival time is announced. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 210 developed model requires, it is suggested that the level of spatial and temporal detail of data for the model should be level C and up. However, an integration of the APC system with the AVL technologies and a radio system is a requirement for the applications of the developed model for real-time bus arrival prediction. The reason is that on one hand, the on-board micro computer can match the vehicle to the route, and on the other hand the stop record can be sent over the air to the control center. 7.8 Summary Chapter 7 is devoted to the methodologies that can predict the likelihood that the bus is on-time, late, early or bunching. These resulted from the conjunction of the real time prediction interval presented in section 7.2 and the historical distribution of bus on- time performance at each bus stop. By knowing in advance the probabilities that the bus is not on-time, bus dispatcher can make a good decision to apply suitable control methods in order to keep service adherence. The broadcasting of real-time bus arrival was also mentioned. As illustrated previously, even if a real-time bus arrival prediction falls within an on-time standard range, there is still a chance that the bus is not on-time and vice-versa since a prediction, in essence, is a probabilistic estimation. The announcement of if the bus is on-time or not should be broadcasted with the consideration of such probabilities. For bus bunching detection, a method which is simple in application was developed. By applying this method, bus dispatchers can easily quantify a likelihood of bus bunching. As a consequence, they will have suitable proactive control methods to prevent it. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 8 CONCLUSIONS AND RECOMMENDATIONS 8.1 Conclusions The growing urban traffic congestion in medium and large size cities is forcing the city authorities to search for efficient solutions in order to reduce the adverse effects. It is widely recognized that among possible solutions, an advanced urban public transit system in the urban areas is potentially the most effective solution. In a modem society where travellers value their time highly, the provision of bus real-time information is instrumental in improving bus transit’s ridership. Therefore, it is important to both public transit providers and the passengers. For transit providers, the development of a real-time bus arrival information system (RETBAIS) is an important objective that they have to meet. For passengers, real-time bus arrival information is an important sign in their perception of a reliable, attractive, and convenient public transit system. Over the last twenty years, many transit providers have been equipping their bus fleets with the AVL and APC systems which are considered to be the technology backbone for the development of a RETBAIS. However, these expensive systems are undemsed. Moreover, the vital role of real-time bus arrival information has been adequately recognized by the transit providers only in the last few years. This thesis research can be considered as a contribution to this new field of bus transit. All thesis research goals and objectives were addressed comprehensively in terms 211 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 212 of methodologies that provide up-to-minute information of bus arrivals as well as to help bus dispatcher the on-line decision support tools for managing the bus fleet. Urban bus transit has a major challenge that it has to strictly adhere to schedule but at the same time it is vulnerable to delays which are very complicated to manage in the urban context. In order to overcome the difficulties of the study of influencing factors on bus arrival time and also to use intensively the available data recoded in the AVL and APC systems, the research separated bus arrival time into two important parts, namely bus running time in a link and bus dwell time at a stop. These philosophies lead to the development of two modules. Following the development of the running time prediction module, based on the basis of a statistical pattern recognition technique, namely the LOWES S method, and the similarities of bus running times experienced in the same period of time in the past, and in the same bus link, a prediction algorithm was formulated. In order to enhance the algorithm’s prediction capability as well as to shorten the prediction time, the author applied leave-one-out cross validation technique and Nadaraya-Watson kernel regression in searching the similarities among variables. The entire algorithm is an automatic computer-intensive procedure for searching, matching and predicting bus running time whenever the AVL-APC data are available and updated. Tests made with simulated and actual data for various bus operation scenarios show that this module performed well, with the average mean relative error were just about 5% for Transitway route and 7% for mixed-traffic bus route. In this type of research, such relative error levels or even higher than these, would be acceptable to the consumer. Moreover, the module worked reliably when dealing with unusual events of bus operation. This characteristic was not found in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 213 the Kalman filter model, a reference model used in this study. It is useful to note that the Kalman filter model outperformed several other comparative models including the Artificial Neural Network-based model in term of bus running time prediction. Following the development of bus dwell time prediction module, based on the logic that the number of boarding passengers and alighting passengers are the determinants for dwell time prediction, the research moved on to the development of a real-time passenger activity prediction algorithm. Four sub-modules were developed separately for predicting boarders, de-boarders and bus dwell time. The fist two sub- modules were also based on the same basis of statistical pattern recognition which was applied in the running time prediction module. In order to avoid using a controversial assumption on the distribution of passenger arrival rate, the algorithm developed for boarding passenger prediction used bus headway and boarding historical data as the elements for a pattern being recognized. It was found that both sub-modules outperformed the reference models and the results were acceptable with the average prediction error of about 22% for boarding prediction, and 6% for alighting prediction. Unlike the first two sub-modules where the predictions were driven by the real time data, the last two sub-modules were developed in the form of pre-defined functions by relating dwell time with the number of boarders, de-boarders, and other selected variables. The first sub-module of these two, namely regression sub-module, was based on the classical functions regressing dwell time to several variables whose data were retrieved form the APC systems. The author found that the regression type A-2.3 is the most suitable with its R-square of up to 63%. In case of data shortage, the type A-2.6 is suggested. The last sub-module, namely the busiest door prediction sub-module, was Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 214 based on the fact that alighting and boarding passengers’ door choice influence dwell time and therefore the determination of the busiest door is important. Several types of binary and multinomial logistic regressions functions, which stem from the discrete choice theory, were examined with the retrieved APC data. The author suggested that binary logistic regression type B-L.l, together with Equations from 5.45 to 5.47, is suitable for rigid-body bus while multiple regression (i.e., type A-2.3) should be used for articulated bus. This developed algorithm also solved the problem of bus stop skipping prediction. When the two modules were integrated for predicting bus arrival times at every bus stop of the full-trips, the entire model worked satisfactorily. The average relative prediction error increased with the new data availability and varied from 3% to 8%. As extensions of the developed model, two algorithms were formulated, one for bus on-time performance evaluation and the other for real-time bus bunching detection. In order to use these algorithms, the author developed a method to delineate real-time bus arrival prediction interval. It is worth noting that most existing bus arrival time prediction models found in the literature could forecast only the mean value of arrival time. Therefore, how certain the prediction is and what the prediction interval is, were not provided by these models. The author of this thesis can quantitatively analyze all measures of bus on-time performance, which again were not found in previous models devoted to bus arrival time prediction. On the basis of the nature of real-time bus arrival predictions, a method to broadcast bus arrivals to passengers was discussed. Bus arrival time broadcasting should be based on the sound probabilities that the bus will be on-time or not. Also, suggestions Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 215 were made on the spatial characteristics and details of the database for the developed model where the integration of the APC system with AVL and a radio system is a prerequisite. This research can be considered as a contribution to the development of real-time bus arrival information system. In this research, the term “ real-time” was illustrated clearly. Here, a real-time prediction model meant to not only to be capable of producing prediction with real-time data but the predictive characteristics of the method itself. Specifically, one important feature for a real-time prediction model is accurate and rapid responsiveness. Therefore, any real-time model should be a computer-intensive one with fast calculation rate. The developed model met this requirement. It was found that a prediction was made by the developed model in only about 15 seconds. All prediction procedures of optimal searching, recognizing, and predicting were done automatically. Hence, this research can provide bus transit providers an accurate, fast, and reliable prediction model. In the case of dwell time prediction, the term “real-time passenger activity prediction” was first introduced by the author and a methodology to predict it was comprehensively developed. The model can help bus transit planner to deal with highly busy bus stops as well as to predict bus dwell time in real-time that should be spent by the buses to serve such stops or to estimate the load factor for each bus, the waiting time of passengers at stop, and so on. Moreover, the algorithm developed in this study can solve a number of existing issues in term of dwell time estimation. By using the methods developed by the author for analyzing bus on-time performance, the bus dispatcher can easily detect the possibilities of “not on-time bus Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 216 arrivals” and “bus bunching”. These capabilities would help transit agencies to provide passengers more accurate and reliable information as well as to manage their bus fleet economically and efficiently. The entire model was developed in a manner that it solely used the AVL-APC data. No additional traffic data such as traffic volume, speed, density or so on, are needed. Moreover, because the data are updated at stops or timepoints (i.e., time-at-location1 ), tracking is not required (Furth et al, 2003). Therefore, the developed model is suitable even if a transit agency is still using the APC systems with the old AVL technologies (i.e., non GPS-based AVL technology). In conclusion, the author believes that the real-time bus arrival time prediction model developed in this thesis will enhance the bus arrival information system and will be a contribution to public transit operation. 8.2 Recommendations and Future Research The model developed in this thesis research based on the statistical pattern recognition technique is innovative and produces results that are clearly more accurate than obtainable from other models. As is the case with other methods, there are limitations of the developed model that should be understood prior to application. However, on the balance, the bus arrival prediction process would be considerably enhanced by adopting the model developed in this thesis. Based on the findings and the 1 Time-at-location: a record of when a bus passed a predetermined point such as a stop or a timepoint. This data collection method is opposite to the location-at-time method - i.e., where a bus was at a given time. Time-at-location is strongly preferred to location-at-time data for management used of achieved AVL-APC data (Furth et al., 2003, p.42). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 217 difficulties experienced during the course of carrying out this research, the following recommendations are made. The results found in chapter 3 raise the question “why the buses running on route 1 are mostly behind the schedule?” and “what should be done to solve that problem?” Based on the distribution of bus on time performance, it was found that the lateness of the buses have been caused not only by the traffic congestion but also by the schedule design. For example, the statistical mean of actual bus running times between Billings Bridge and Gladstone stops is about 15 minutes whereas the running time scheduled for the buses of this segment is only 11 minutes. With that set up, the probability for the bus “on-time” was only about 43%. Therefore, the OC transpo should adjust the scheduled running times between stops closer to the actual means. Such an adjustment would be appreciated by their customers. In chapter 4, the author used Euclidian distance to calculate the similarity among patterns. Mathematically speaking, the author actually treated the elements in a pattern matrix equally with the same weights. However, running times of the buses closer to the understudied bus may get more weights because they run under the most similar road and traffic condition with the bus of interest. Further study should be carried on with this standpoint in order to improve the developed model. Due to constraints in data collection and simulation, a number of elements taken into a pattern in this study were eliminated in order to ensure a sound statistical inference. However, whenever the database is large, more elements should be added into the patterns so that it can improve the prediction results of the developed model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 218 The elements in the form of categorical variables should be considered. For example, for a pattern of boarding passenger prediction, besides the elements which were presented, we can add categorical elements such as bus type, seating plan, and so on. The method to recognize such patterns should be further explored in order to make the developed model more versatile. It is possible that matrix B = (XTWX) in equation 4.7 will become a singular matrix so that no prediction can be made. In order to solve the problem, further study to convert the linear regression shown in chapter 4 into the form of ridge regression is needed. Finally, data organization for the retrieved APC-AVL data as well as the data provisions for each module of the developed model requires further studies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES Abdelfattah, AliM., and Khan Ata M., (1998). Models for Predicting Bus Delays. Transportation Research Record 1623, pp. 8-15. Abkowitz, Mark, Eiger Amir, and Engestein Israel, (1986). Optimal Control of Headway Variation on Transit Routes. Journal o f Advanced Transportation, Vol.20 (1), p p . 73-88. Abkowitz, Mark, and Lepofsky, (1988). Implementing Headway Based Reliability Control on Transit Routes. Journal of Transportation Engineering, Vol. 116, No.l, pp. 49- 63. A C Transit. Rider Information, http://www.actransit.org Akcelik & Associates Pty Ltd. Micro-Simulation and Analytical Methods for Modeling Urban Traffic. October 2001. Akcelic, R., and Besly M., (2001). Acceleration and Deceleration Models. The 23rd Conference of Australian Institutes of Transport Research (CAITR 2001). Monash University, Melbourne, Australia, December 2001. Alfa, Attahiru Sule, William B.Menzies, James Purcha, and R.McPherson, (1988). A regression Models for Bus Running Times in Suburban Areas of Winnipeg. Journal o f Advanced Transportation, Vol. 21, pp. 221-221. Andersen, Bob, (2005). Introducing Nonparametric Regression. Lecture notes. McGill University. 219 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 220 Ben, Akiva, and Steven Lermark, (1985). Discrete Choice Analysis. The MIT Press Bertini, R.L., and Ahmed M., Geneidy El., (2004). Modeling Transit Time Using Archived Bus Dispatch System Data. Journal of Transportation Engineering, ASCE Vol. 130, No. 1, January 1. Berry, W., Feldman S., (1985). Multiple Regressions in Practice. Sage Publications. Bodin, L.D., Ball M.O., Duguid R., and Mitchell M., (1985). The Vehicle Scheduling Problem with Interlining. Computer scheduling of Public Transport 2, Edited by Jean-Marc Rousseau, Elsevier Science Publishers B.V, pp.427-450. Chang, Hyeon, Theo Arentze, and Harry Timmermans, (2001). Pattern Recognition in Complex Activity Travel Patterns; Comparison of Euclidian Distance, Signal- Processing, Theoretical and Multidimensional Sequence Alignment Methods. Transportation Research Record 1752, pp. 16-22. Chapman, R.A., and Michel J.F, (1978). Modeling the Tendency of Buses to Form Pair. Transportation Science, Vol. 12, No. 2, pp. 165-175. Transportation Research record 1752, pp. 16-22. Cathey, F.W., and D.J. Dailey, (2003). A Prescription for Transit Arrival/ Departure Prediction Using Automatic Vehicle Location Data. Transportation Research Part C, V ol.ll,pp.241-264. Chen, Mei, Liu Xiaobo, Xia Jingxin, and Chien Steven, (2004). A Dynamic Bus- Arrival Time Prediction Model Based on APC Data. Computer- Aided Civil and Infrastructure Engineering, Vol. 19, pp. 364-376. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 221 Chien, Steven I., Shoai M. Chowdhury, Kyriacos C., Mouskos, and Yuqing Ding, (2000). Enhancements of CORSIM Model in Simulating Transit Operations. Journal o f Transportation Engineering, Vol. 126, No. 5, September 1, pp. 396-404. Chien, Steven I., Yuqing Ding, and Chienhung Wei., (2002). Dynamic Bus Arrival Time Prediction with Artificial Neural Networks. Journal o f Transportation Engineering, Vol. 128, No.5, September 1, pp. 429-438. Cleveland, William S., (1979). Robust Locally Weighted Regression and Smoothing Scatter Plots. Journal of the American Statistical Association, Vol.74, No.368, pp.829-836. Cleveland, William S., (1988). Locally Weighted Regression: An Approach by Local Fitting. Journal o f the American Statistical Association, Vol. 88, pp. 596-610. Conover, W.J., (1971). Practical Nonparametric Statistics. John Wiley &Sons Inc. Dailey, D.J, S.D Maclean, F.W. Cathey, and Z.R. Wall, (2001). Transit Vehicle Arrival Prediction, Algorithm and Large-Scale Implementation. Transportation Research Record, 1771, pp. 46-51. Daniel, L. Gerlough and Frank C. Barnes., (1971). Poisson and Other Distributions in Traffic. Eno Foundation for Transportation. Saugaruck, Conecticut. Daniel, Wayne W., (1978). Applied Nonparametric Statistics. Houghton Mifflin Company. Daniel, K. Boley, (1998). Passenger Counting Technologies and Procedures, TCRP synthesis 29, TRB, Washington, DC. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 222 Davis, Gary A., Nacy L.Nihan, Mohammad M.Hamed, and Lesli N. Jacobson, (1990). Adaptive Forecasting of Freeway Traffic Congestion. Transportation Research Record, 1287, pp. 29-33. Devroye, Luc, Laszlo Gyorfi, and Lugosi, (1996). A probabilistic Theory of Pattern Recognition. Springer Publication. Dowling, R., Skabardonis A., and Alexiadis V., (2004). Traffic Analysis Toolbox- Volume 3: Guidelines for Applying Traffic Simulation Software. Report FHWA- HRT-04-040. Ding, Yuquing, and Chien Steven, (2002). Improving Transit Quality and Headway Regularity with Real-Time Control. Transportation Research Record, 1760, pp. 161-170. Dueker, Kenneth J., Thomas J. Kompel, and James G. Strathman, (2004). Determinants of Bus Dwell Time. Journal of Public Transportation, Vol.7, No.l, pp. 21-38. Website http://www.nctr.usf.edu/ipt/ioumalarticles.html Ernest, Bellman R., (1961). Adaptive Control Process. Princeton, N.J, Princeton University Press. Fan and Gijbels I., (1996). Local Polynomial Modeling and Its Applications. Chapman & Hall Publisher. Fang, Zhao and Min-Tang-Li., (2005). Calibration of Highway / Transit Speed relation For Improved Transit network Modeling in FSUTMS. BD-015-07, Final Report. Lehman Center for Transportation Research - Florida International University. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 223 Fang, Zhao and Nokil Park, (2004). Using Geographically Weighted Regression Models to Estimate Annual Average Daily Traffic. Transportation Research Record, 1879, pp. 99-107. Farhan, Ali, (2002). Bus Arrival Time Prediction for Dynamic Operations Control and Passengers Information Systems. Thesis Dissertation of Masters of Applied Science, University of Toronto, Canada. Fox, John, (1991). Regression Diagnostics. Series in Quantitative Applications in Social Sciences, Sage University Paper, Vol.79. Fox, K., (2000). SMARTEST-new tools for evaluating ITS. Traffic Engineering and Control, Vol.41 (1), pp.20-22. Fukotomi, Akihira, (2004). Simulation Approach to Study o f Drving Behaviour, Fuel Consumption and Emissions. Master Thesis Dissertation, Carleton University, Ottawa-Canada. Fu, Liping and Liu Qing, (2004). Real-time Optimization Model for Dynamic Scheduling of Transit Operations. Transportation Research Record, 1857, pp. 48- 55. Fu, Liping and Yang X., (2003). Design and Implementation of Bus-Holding Control Strategies with Real-time Information. Transportation Research Record, 1791, pp. 6- 12. Furth, G. Peter, (2003). Use of Archived AVL-APC Data to Improve Transit Performance and Management: Review and Potential. Transit Cooperative Research Program Web Document 23 (Project H-28), TRB, Washington, DC. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 224 Gibbons, J. Dickinson., (1984). Nonparametric Methods for Quantitative Analysis. Second Edition. American Sciences Press, Inc. Guether Richard P., and Kumares C. Sinha, (1983). Modeling Bus Delays due to Passenger Boarding and Alighting. Transportation Research Record, 915, pp. 7-13. Gyorfi Lazio, Hardle Wolfgang, Sarda Pascal, and Vieu Philippe, (1989). Nonparametric Curve Estimation from Time Series. Springer-Verlag. Harris, Richard, (2000). A Primer of Multivariate Statistics. Third Edition, IEA Publisher Hardle, Wolfgang, (1990). Applied Nonparametric Regression. Cambridge University Press. Hay, William and Winkler Robert, (1971). Statistics, Probability, Inference, and Decision. Holt, Rinehart and Winston, Inc. Helali, K.N and Khan A.M.., (1993). Regression Based Urban Arterial Road Delay Models. Canadian Journal of Civil Engineering, Vol. 20, No.l, Feb. 1993, pp.37- 49. Higham, Desmond and Higham Nicholas, (2000). MATLAB Guide. SIAM Philadelphia. Horbury, Antoneta X., (1999). Using non-real-time Automatic Vehicle Location Data to Improve bus Services. Transportation research Part B. V33. pp. 559-579. Ishak, S, C., and Alecsandru, (2004). Optimizing Traffic Prediction Performance of Neural Networks under Various Topological, Input, and Traffic Condition Settings. Journal o f Transportation Engineering, Vol. 130, No.4, July 1, pp. 452-465. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 225 Iversen, G. R, and Norpoth Helmut, (1976). Analysis o f Variance. Sage Publications. Kalaputapu, R., and Demetsky, (1985). Application of Artificial Neural Networks and Automatic Vehicle Location Data for Bus Transit schedule behavior Modeling. Transportation Research Record 1497, pp.44-52. Kimpe, J. Thomas., James G. Strathman, David Griffin, Steven Callas, and Richard L. Gehart, (2003). Automatic Passenger Counter Evaluation Implication for National Transit Database Reporting. Transportation Research Record 1835, pp. 93-99. Kittelson, and Associates, Inc., (2003). Transit Capacity and Quality o f Transit Manual. The Second Edition. TRB, Washington D.C Klopfenstein, R., (1986). “Bus Communication System”. Transportation Research Board, National Research Council. Washington D.C. Koutsopoulos, H.N, A.R. Odoni, and N.H.M Wilson, (1985). Determination of Headway as Function Time varying Characteristics on a Transit Network. Computer scheduling of Public Transport 2, Edited by Jean-Marc Rousseau, Elsevier Science Publishers B.V., 1985. pp. 391-413. Kraft, W., and TH. Deutschman, (1974). Bus Passenger Service-time Distribution. Transportation Research Record, 625, pp.37-43. Kwon, Coifman B., and Bickel P., (2000). Day to Day Travel Time Trends and Travel Time Prediction from Loop Detector Data. Transportation Research Record, 1717, pp. 120-128. Leuzbatch, Wilhelm, (1988). Introduction to the Theory o f Traffic Flow. Springer Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 226 Levine, Jonathan C., and Gwo-Wei Torng, (1994). Dwell-time Effects of Low-Floor Bus Design. Journal o f Transportation Engineering, Vol. 120, No.6, November, pp. 914-929. Levinson, Herbert, (1983). Analyzing Transit Travel Time Performance. Transportation Research Record, 915, pp. 1-7. Li, Weigang, Marlon W. Koendjbiharie, Ricardo C.deM.JuCa’, Yaeko Yamashita, and Andrew Maciver, (?). Algorithms for Estimating Bus Arrival Times Using GPS Data. Retrieved from the Website Linda, B., and Virginia C., (1992). Processing data-The survey Example. Series Quantitative Applications in Social Sciences, Sage University Paper 85. Lewis P.A, and Orav. E.J.,(1980). Simulation Methodology for Statistic, Operation and Analysts, Engineers -Volume 1. Wadsworth & Brooks/code Statistics/ Probability Series. Lin, T., and N.H. Wilson, (1992). Dwell time Relationships for Light Rail Systems. Transportation Research Record, 1361, pp. 287-295. Lin ,Wei Hua and Bertini Rober L., (2004). Modeling Schedule Recovery Process in Transit Operations for Bus Arrival Time Prediction. Journal of Advanced Transportation, Vol. 38, No.3, pp. 347-365. Lin, Wei-Hua and Zeng Jian, (1999). Experimental Study of Real- Time Bus Arrival Time Prediction with GPS Data. Transportation Research Record, 1666, pp. 101- 109. Loader, Clive, (1999). Local Regression and Likelihood. Statistics and Computing, Springer. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 227 Lyshevski, Sergey, (2003). Engineering and Scientific Computations Using MATLAB, John Wiley and Sons. Maloney, Mark and Boyle Daniel, (1999). Components of Travel Time on the Glendale Beeline Bus Network. Transportation Research Record 1666, pp. 23-27. Marshall, Leo F., Herbert S. Levison, Lawrence C. Lennon, and Jerry Cheng, (1990). Bus Service Times and Capacities in Manhattan. Transportation Research Record, 1266, pp. 189-196. Maybeck, P.S.,(1979). Stochastic Models, Estimation, and Control. Volume 1. Academic Press. McKnight, Claire, Herbert Levison, Kaan Ozbay, Cammile Kamga, and Rober E. Passwell, (2003). Impact of Congestion on Bus Operations and Costs. FHWA-AV- 2003-008, Final Report. McKnight C., and Passwell R.E, (1997). Impact of Congestion on New York Bus Service. UTRCfor MTA NewYork City Transit. Mei, Chen and Chien Steven I. J., (2001), Dynamic Freeway Travel-Time Prediction with Probe Vehicle Data. Transportation Research Record, 1768, pp. 157-161. Menard, Scott, (2001). Applied Logistic Regression Analysis. 2sd Edition, Sage University Paper. Series: Quantitative Applications in the Social Science. Mishalani, Rabi G., Sungjoon, Lee and Mark, R. McCord, (2000). Evaluating Real- Time Bus Arrival Information Systems. Transportation Research Record, 1731, pp. 81-87. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 228 Morgan, D., (2002). A Microscopic Simulation Laboratory for Advanced Public Transportation System Evaluation. Master Thesis Dissertation, MIT. Neter J., Wasserman W., and Kutner M., (1985). Applied Linear Statistical Models. Second Edition, Richard D.IrWIN, Inc. Normand R., Draper and Harry, Smith, (1998). Applied Regression Analysis. Wiley series in Probability and Statistics. Wiley Inter-science Publishers. Third Edition. Okunieff, Paula E., (1997). AVL Systems for bus Transit. Synthesis o f Transit Practice 24, TRR, National Academy Press, Washington D.C Patnaik, J., Steven Chien, and Athanassios, Bladikas, (2004). Estimation of Bus Arrival Times Using APC Data. Journal of Public Transportation, Vol.7, No.l, pp. 1-19. Website: http://www.nctr.usf.edu/ipt/ioumalarticles.html Pile, Randall, Jame Niemayer, and Rusell Chrishom, (1998). Transit Scheduling: Basic and Advanced Manuals. TCRP Report 30, TRB, Washington, DC. Pedhazur, Elazar, (1997). Multiple Regression in Behavioral Research, Explanation and Prediction. Harcourt College Publishers, Third Edition. PTV America, (2004). VISSIM User Manual. PTV Planung Transport Verkehr AG PTV America, (2005). NEMA Editor Manual (Version3.3). PTV Planung Transport Verkehr AG Rabi, Mishalani, Sungjoon Lee, and Mark R.McCord., (2000). Evaluating Real-time Bus Arrival Information Systems. Transportation Research Record 1731, pp. 81- 87. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 229 Rajbhandari Rajat, Chien Steven I., and Daniel Janice R., (2003). Estimation of Bus Dwell Times with Automatic Passenger Counter Information. Transportation Research Record, 1841, pp. 120-127. Reinhoundt, Edwin M., and Veslatin S.A., (1997). A Dynamic Predicting Algorithm for Estimating Bus Arrival Time. Presents of the 8th IFAC/IFIP/IFORS Symposium, Crete, Greece, June 1997. Vol3. pp. 1225-8. Richard.P., and Dennis C.,(1984). Cross-validation of Regression Models. Journal o f the American Statistical Association, Vol.79, No.387, pp.575-583. Roess, Roger, Prassas Elena, and McShane William., (2004). Traffic Engineering, 3rd Edition, Prentice Hall. Ruihong, Huang and Zhong-Ren Peng., (2002). Schedule-Based Path-Finding Algorithms for Transit Trip-Planning Systems. Transportation Research Record, 1783, pp. 142-148. Schalkoff, Robert, (1992). Pattern Recognition, statistical, structural and neural Approaches. John Wiley & Sons, Inc. Schiavon, John, (1999). Understanding and Applying Advanced on Board Bus Electronics. TCRP Report 43. Transportation Research Board. Schuldze, Thomas, and Fliees Thomas., (1997). Urban Traffic Simulation with Psycho physical Vehicle-Following Models. Proceedings o f the 1997 Winter Simulation Conference. Schweiger, L., (2003). Real-time Bus Arrival Information Systems, a Synthesis of Transit Practice. Transportation Research Board. Washington D.C Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 230 Senevirante, P.N., (1988). Simulation of Fixed Bus Travel Time. Journal of Advanced Transportation, Vol. 22, pp. 39-53. Senevirate, Prianka, (1990). Analysis of On-time Performance of Bus Services Using Simulation. Journal o f Transportation Engineering, Vol.116, No.4, pp. 517-531. Shalaby, A., Farhan Ali., (2004). Prediction Models of Bus Arrival and Departure Times Using AVL and APC Data. Journal of Public Transportation, Vol.7, No.l, pp. 41-60. Website: http://www.nctr.usf.edu/ipt/ioumalarticles.html Shara R.J., (2002).Development o f a Public Transit Information System Using GIS and ITS Technologies. Master Thesis Dissertation, Carleton University, Ottawa- Canada. Silverman, B.W., (1986). Density Estimation for Statistic and Data Analysis. Monographs on Statistics and Applied Probability. Skomal, Edward, (1981). Automatic Vehicle Locating System. Van Nostrand Reihold Company. Smith, Brian L., and Michael J., Demetsky, (1994). Short-term Traffic Flow Prediction: Neural Network Approach. Transportation Research Record, 1453, pp. 98-104. Son, Bongsoon, Kim Hyung Jin, Shin Chi huyn, and Lee sang keon, (2004). Intelligent Multimedia Solusion and security for the next generation Mobile Network-Bus Arrival Time Prediction Method for ITS Application. Lecture Notes in Computer Science. V.3215, pp.88-94. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 231 Stephan, R.S., Kenth A.B, and David W.S., (1994). Cross-validation of Multivariate Densities. Journal o f the American Statistical Association, Vol.89, No.427, pp.807- 817. Strathman, Janies G., Thomas J.Kimple, and Kentheth J. Dueker,(2000). Bus Transit Operations Control: Review and Experiment Involving Tri-Met’s Automated Bus Dispatching System. Sun, HongYu, Liu Henrry, Xiao Heng, He Rachel, and Ran Bin, (2003). Use of Local Linear Regression Model for Short-term Traffic Forecasting. Transportation Research Record, 1836, pp. 143-150. Transportation research Board. Highway Capacity Manual. Special Report 209. TRB, National Research Council, Washington, D.C, 2000. Transit Cooperative Research Program, (2001). Transit Capacity and Quality of Service Manual, Second Edition, TRB, Washington, DC. Transit Cooperative Research Program, (2000). Strategies for Improved Traveler Information, TCRP Report 92, TRB, Washington, DC. Transit Cooperative Research Program, (1998). Understanding and Applying Advanced On-Board Bus Electronics, Report 43, TRB, Washington, DC. Turnquist, MA., (1982). Strategies for Improving Bud Transit Service Reliability. Evanston, IL: Northwestern University. (NTIS Report No. DOT/RSPA/DPB-50-81- 27).USDOT, Research and Special Programs Administration. Ubitran and Levison, (1986). Bus Priority Improvement Study : Analysis O f bus Priority Proposal. November 1986 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 232 Versteegrt, H.H, and Temprere C., MJ., (2003). PredicTime, a Stage of the Art and Functional Architecture. TNO- Inro Report 2003-07. Institute for Traffic and Transport, Logistic and Spatial Development, Nederland. Vuchic, VuKan R., (2004). Urban Transit Operation, Planning, and Economics. John Willey and Sons, Inc. Webb, Andrew, (2002). Statistical Pattern Recognition. John Wiley and Sons Ltd. Second Edition. Wei, Hua Lin, Qingying Lu, and Joy Dahgren, (2002).Dynamic Procedure for Short- Term Prediction of Traffic Conditions. Transportation Research Record, 1783, pp. 149-157. Wei, Hua Lin, Amit Kulkarni, and Pitu Mirchandani, (2004). Short-term Arterial Travel Time Prediction for Advanced Traveler Information Systems. Intelligent Transportation System, Vol.8, pp. 143-154. Weidemann, Reiter U., (1994). Micro-traffic simulation, the simulation system mission: Background and Actual State. Publication (no information). Weigang, Li, Koendjbiharie Marlon, Ricardo C.de M.JuCa, Yaeko Yamashita, and Andrew Macive, (2004). Algorithm for Estimating Bus Arrival Times using GPS data. Retrieved from Website. William, R. Schucany, (1995). Adaptive Bandwidth Choice for Kernel Regression. Journal o f American Statistical Association, Vol.90, No.430, pp.535-540. Zargari, S., (1997). Optimization o f Integrated Multimodal Urban Transportation Corridors. Ph.D Thesis Dissertation, Carleton University, Ottawa, Canada. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 233 Zargari S., and Khan A.M., (2003). Optimization of Travel in Bus Rapid Transit-Based Multimodal Corridors. Journal o f Public Transportation, Vol.6, No.2, pp.87-104. Zhang, Wenyang and Lee Sik-Yum, (2000). Variable Bandwidth Selection in Varying- Coefficient Models. Journal o f Multivariate Analysis, Vol. 74, pp. 116-134. Zhong, Ren Peng, Danlin Yu, and Edward Beimborn., (2002). Transit User Perceptions of the Benefits of Automatic Vehicle Location. Transportation Research Record, 1791, pp. 127-133. Zong, Z., Tian, Thomas U., Roelof E., and Kevin B., (2002). Variations in Capacity and Delay Estimates from Microscopic Traffic Simulation Models. Transportation Research Record, 1802, pp. 23-31. Yoon, Paul and Hwang, Ching-lai, (1995). Multiple Attribute Decision Making. An Introduction series Quantitative Applications in Social Sciences, Sage University Paper 104. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDICIES 234 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix A 1. General This appendix shows the maps of bus route 95 and route 1. The maps were retrieved from the Website www.octranspo.com 2. Bus Route Maps OI I nmm*iMu Orleans Fallowfield SMwee* d» f74. Sonricsaner CCaoroi St )HM * «Ht»cw«.rla IW9 . gABtf MORNING SERVICE % SERVICE M tftNFato«rtMW*Rac«tfOr&3nc A L aurtft __ mbm* rrj] yssraaMCow^eeoo 1 TtMpaMlHtumtOtp Figure A.l: Bus route 95 235 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 236 \9um usm South Keys 5Rf '« Ottawa-Rockcliffe toft* I^LttiSOOWWEPARK CWtffSSg*®* ftMfcsuCaretf flMfcsffirisau - Riverside * ^ 0 - -mmt swEnwiafjjJj nmon ggf feg Tti^Kvtyt Stated AUi*, EARLY MORNING SERVICE befcra 6 *m iwfcewn W m z gftewbofp A Wtfou Cw>t 5iggy*CCMArftWi.»>awl M en tis O nonboroei te 4 TiRxpirt/ NpeMW^pi$»9e Figure A.2: Bus route 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix B1 1. General This Appendix covers the results of six regression types described in chapter 5. 2. SPSS results for Classical Regression Types Table B l:l Model Summary for Type A-2.1 Model Summary1 Adjusted Std. Error of Durbin- Model R R Square R Square the Estimate Watson 1 .781a .610 .609 7.35328 2 .781b .610 .609 7.35280 o 3 00 .610 .609 7.35339 1.800 a- Predictors: (Constant), DOORS, SEASON, PUNT, TotalOnPw, TOTAL OFFS, STOP_LOCATE, LF, TOTAL_ONS, LOAD_ARR b- Predictors: (Constant), DOORS, SEASON, PUNT, TotalOnPw, TOTAL OFFS, STOP_LOCATE, LF, TOTAL_ONS c. Predictors: (Constant), DOORS, PUNT, TotalOnPw, TOTAL_OFFS, STOP LOCATE, LF, TOTAL_ONS d- Dependent Variable: DWELL_3 237 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 238 Table B1.2: Parameters of Type A-2.1 Coefficients Unstandardized Standardized Coefficients Coefficients Cdlinearih Statistics Model B Std. Error Betat SiQ. Tolerance V1F 1 (Constant) 13.346 1.070 12.469 .000 TOTALOFFS .438 .015 .250 28.700 .000 .843 1.186 TotalOnPw -.030 .001 -.413 -20.587 .000 .159 6.287 TOTALONS 1.928 .037 1.054 51.417 .000 .152 6.558 LOADARR .012 .027 .019 .450 .653 .034 29.251 LF -1.532 1.575 -.039 -.973 .331 .040 25.161 PUMT .003 .001 .048 5.441 .000 .840 1.191 SEASON .118 .084 .011 1.414 .158 .990 1.010 STOP_LOCATE 1.975 .206 .083 9.573 ,000 .846 1.182 DOORS -4.050 .399 -.150 -10.152 .000 .295 3.386 2 (Constant) 12.961 .645 20.095 .000 TOTALOFFS .439 .015 .251 28.871 .000 .851 1.176 TotalOnPw -.030 .001 -.414 -20.673 .000 ,160 6.255 TOTALONS 1.929 .037 1.055 51.657 .000 .154 6.508 LF -.840 .344 -.021 -2.443 .015 .833 1.200 PUNT .003 .001 .047 5.437 .000 .840 1.191 SEASON .118 .084 .011 1.410 .158 .990 1.010 STOPLOCATE 1.956 .202 .083 9.682 .000 .882 1.134 DOORS -3.902 .227 -.144 -17.219 .000 .915 1.093 3 (Constant) 13.177 .627 21.024 .000 TOTALOFFS .438 .015 .250 28.834 .000 .853 1.173 TotalOnPw -.030 .001 -.413 -20.636 .000 .160 6.248 TOTALONS 1.928 .037 1.054 51.635 .000 .154 6.500 LF -.830 .344 -.021 -2.414 .016 .834 1.199 PUNT .003 .001 .048 5.456 .000 .840 1.191 STOPLOCATE 1.957 .202 .083 9.684 .000 .882 1.134 DOORS -3.922 .226 -.145 -17.342 .000 .919 1.089 a- Dependent Variable: DWELL_3 Table B1.3: Model Summary for Type A-2.2 Model Summar/ Adjusted Std. Error of Durbin- Model R R Square R Square the Estimate Watson 1 .786a .618 .618 7.26409 2 .786b .618 .618 7.26403 1.843 a. Predictors: (Constant), TotalOnCubic, PUNT, SEASON, DOORS, TOTAL OFFS, STOP_LOCATE, LF, TOTAL_ONS, LOAD_ARR, TotalOnPw b. Predictors: (Constant), TotalOnCubic, PUNT, SEASON, DOORS, TOTAL OFFS, STOP_LOCATE, LF, TOTAL_ONS, TotalOnPw c- Dependent Variable: DWELL_3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 239 Table B1.4: Parameters of Type A-2.2 Coefficients3 Unstandardized Standardized Coefficients Coefficients Collinearitv Statistics Model B Std. Error Betat Sig. Tolerance VIF 1 (Constant) 13.664 1.075 12.706 .000 TOTAL_OFFS .427 .015 .243 27.949 .000 .834 1.200 TotalOnPw -.071 .006 -.964 -11.837 .000 .010 104.957 TOTAL_ONS 2.347 .070 1.288 33.545 .000 .043 23.319 LOAD_ARR .026 .027 .041 .956 .339 .034 29.035 LF -2.415 1.566 -.061 -1.542 .123 .040 25.093 PUNT .003 .001 .048 5.485 .000 .831 1.203 SEASON .153 .084 .015 1.824 .068 .985 1.015 STOP_LOCATE 1.848 .205 .078 8.995 .000 .839 1.192 DOORS -4.387 .399 -.161 -10.989 .000 .295 3.392 TotalOnCubic .001 .000 .361 7.034 .000 .024 41.712 2 (Constant) 12.843 .649 19.804 .000 TOTALJDFFS .429 .015 .244 28.173 .000 .841 1.189 TotalOnPw -.071 .006 -.966 -11.873 .000 .010 104.851 TOTALJDNS 2.351 .070 1.290 33.657 .000 .043 23.241 LF -.954 .344 -.024 -2.771 .006 .823 1.214 PUNT .003 .001 .048 5.490 .000 .831 1.203 SEASON .153 .084 .015 1.821 .069 .985 1.015 ST OP_LOCATE 1.808 .201 .076 8.987 .000 .874 1.144 DOORS -4.073 .227 -.149 -17.953 .000 .913 1.095 TotalOnCubic .001 .000 .362 7.050 .000 .024 41.700 a - Dependent Variable: DWELL_3 Table B1.5: Parameters of Type A-2.4 Parameter Estimates 95% Confidence Interval Parameter Estimate Std. Error Lower Bound Upper Bound b1 31.287 .384 30.534 32.039 b2 .346 .014 .319 .374 b3 .795 .005 .784 .805 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 240 Table B1.6: Model Summary for Type A-2.4 ANOVA? Sum of Mean Source Squares df Squares Regression 1627451 3 542483.7 Residual 405012.0 6032 67.144 Uncorrected Total 2032463 6035 Corrected Total 813235.7 6034 Dependent variable: DWELL_3 a. R squared = 1 - (Residual Sum of Squares) / (Corrected Sum of Squares) = .502. Table B1.7: Parameters of Type A-2.5 Parameter Estimates 95% Confidence Interval Parameter Estimate Std. Error Lower Bound Upper Bound b1 1184.802 1268661 -2485838.131 2488207.734 b2 91.856 998524.5 -1957368.933 1957552.645 b3 -2.557 9811.205 -19235.985 19230.870 b4 -.003 7.131 -13.981 13.976 Table B1.8: Model Summary for Type A-2.5 ANOVA? Sum of Mean Source Squares df Squares Regression 1643479 4 410869.8 Residual 441527.8 6095 72.441 Uncorrected Total 2085007 6099 Corrected Total 843898.3 6098 Dependent variable: DWELL_3 a- R squared = 1 - (Residual Sum of Squares) / (Corrected Sum of Squares) = .477. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 241 Table B1.9: Parameters of Type A-2.6 Parameter Estimates 95% Confidence Interval Parameter Estimate Std. Error Lower Bound Upper Bound b1 -1.496 1.613 -4.658 1.666 b2 97.767 234.859 -362.640 558.175 b3 470.907 1253.074 -1985.570 2927.383 b4 .561 .117 .332 .790 Table B1.10: Model Summary for Type A-2.6 ANOVA? Sum of Mean Source Squares df Squares Regression 1643120 4 410779.9 Residual 387654.5 5996 64.652 Uncorrected Total 2030774 6000 Corrected Total 819684.1 5999 Dependent variable: DWELL_3 a- R squared = 1 - (Residual Sum of Squares) / (Corrected Sum of Squares) = .527. Table B l.ll: Model Summary for Pilot Regression without Deletion of Outlier and Extreme values Model Summai$ Chanqe Statistics A djusted Std. Error of R Square Durbin- Model R R S q u are R S q u are the Estimate C hanqe F C hange df1 df2 Siq. F Change W atson 1 .608a .370 .364 15.99027 .370 65.929 8 898 .000 2 ,608b .370 .365 15.98183 .000 .052 1 898 .820 3 .607° .369 .365 15.98773 -.001 1.664 1 899 .197 2.041 a. Predictors: (Constant), BUS_TYPE, STOP_LOCATE, SEASON, TOTAL_ONS, PUNT, LF, TOTAL_OFFS, LOAD_DEP b. Predictors: (Constant), BUS_TYPE, STOP_LOCATE, SEASON, TOTAL_ONS, LF, TOTAL_OFFS, LOAD_DEP c- Predictors: (Constant), BUS_TYPE, STOP_LOCATE, TOTAL_ONS, LF, TOTAL_OFFS, LOAD_DEP b. Dependent Variable: DWELL_1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 242 Table B1.12: Coefficients for the Pilot Regression Coefficients? Unstandardized Standardized Coefficients Coefficients 95% Confidence Interval for B Collinearitv Statistics Model B Std. Error Beta t Sifj. Lower Bound Upper Bound Tolerance VIF 1 (Constant) 17.323 1.958 8.846 .000 13.480 21.166 STOP_LOCATE 1.819 1.182 .044 1.539 .124 -.501 4.139 .843 1.187 SEASON .605 .468 .034 1.292 .197 -.314 1.524 .989 1.011 PUNT .001 .003 .007 .228 .820 -.006 .007 .835 1.198 LF -27.184 9.753 -.409 -2.787 .005 -46.325 -8.044 .033 30.758 TOTAL_OFFS .829 .178 .280 4.643 .000 .478 1.179 .193 5.176 LOAD_DEP .408 .165 .392 2.467 .014 .083 .733 .028 35.896 TOTAL_ONS 1.124 .189 .381 5.936 .000 .753 1.496 .170 5.876 BUS_TYPE -15.769 2.268 -.328 -6.952 .000 -20.221 -11.317 .315 3.170 2 (Constant) 17.314 1.957 8.848 .000 13.473 21.154 STOP_LOCATE 1.879 1.151 .046 1.632 .103 -.380 4.139 .887 1.127 SEASON .604 .468 .034 1.290 .197 -.315 1.522 .989 1.011 LF -27.081 9.737 -.408 -2.781 .006 -46.191 -7.971 .033 30.692 TOTAL_OFFS .831 .178 .280 4.661 .000 .481 1.180 .194 5.166 LOAD_DEP .408 .165 .392 2.470 .014 .084 .733 .028 35.895 TOTAL_ONS 1.125 .189 .381 5.945 .000 .754 1.497 .170 5.874 BUS_TYPE -15.810 2.260 -.329 -6.996 .000 -20.245 -11.374 .317 3.151 3 (Constant) 18.216 1.828 9.963 .000 14.627 21.804 ST 0 P_LOCATE 1.923 1.151 .047 1.671 .095 -.336 4.183 .888 1.126 LF -27.223 9.740 -.410 -2.795 .005 -46.338 -8.107 .033 30.688 TOTAL_OFFS .829 .178 .280 4.649 .000 .479 1.179 .194 5.165 LOAD_DEP .410 .165 .393 2.479 .013 .085 .734 .028 35.893 TOTALJDNS 1.128 .189 .382 5.957 .000 .756 1.500 .170 5.873 BUS_TYPE -15.965 2.258 -.332 -7.072 .000 -20.396 -11.535 .318 3.142 a- Dependent Variable: DWELL_1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix B2 1. General This appendix presents the outputs of SPSS 13 applied for three types of binary logistic regressions. Each type has three tables recording the model summary, the omnibus tests, and model parameters. 2. SPSS Results for Binary Logistic Regression Table B2.1: Model Summary for Type B-L.l Model Summary -2 Log Cox & Snell Nagelkerke Step likelihood R Square R Square 1 1804.2293 .418 .586 2 1804.7773 .418 .586 a. Estimation terminated at iteration number 8 because parameter estimates changed by less than .001. Table B2.2: Omnibus Tests of Model Coefficients of Type B-L.l Omnibus Tests of Model Coefficients Chi-square df Siq. Step 1 Step 1379.104 10 .000 Block 1379.104 10 .000 Model 1379.104 10 .000 Step 2s Step -.548 2 .760 Block 1378.557 8 .000 Model 1378.557 8 .000 a. A negative Chi-squares value indicates that the Chi-squares value has decreased from the previous step. 243 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 244 Table B2.3: Model Parameters of Type B-L.l Classification Tabid* Predicted BUSIER DOOR Busier door Busier Door Percentage Observed is rear door is Front door Correct Step 1 BUSIER_DOOR Busier door is rear door 647 161 80.1 Busier Door is Front door 189 1551 89.1 Overall Percentage 86.3 Step 2 BUSIER_DOOR Busier door is rear door 647 161 80.1 Busier Door is Front door 188 1552 89.2 Overall Percentage 86.3 a- The cut value is .500 Table B2.4: Model Summary for Type B-L.2 Model Summary -2 Log Cox & Snell Nagelkerke Step likelihood R Square R Square 1 942.654® .262 .473 2 942.743® .262 .473 3 944.813b .261 .471 4 948.127° .260 .469 5 948.378d .260 .469 a- Estimation terminated at iteration number 15 because parameter estimates changed by less than .001. b- Estimation terminated at iteration number 16 because parameter estimates changed by less than .001. c. Estimation terminated at iteration number 10 because parameter estimates changed by less than .001. d. Estimation terminated at iteration number 8 because parameter estimates changed by less than .001. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 245 Table B2.5: Model Parameters of Type B-L.2 Classification Tabl£ Predicted BUSIER DOOR Busier door Busier Door Percentage Observed is rear door is Front door Correct Step 1 BUSIER DOOR Busier door is rear door 102 159 39.1 Busier Door is Front door 41 1566 97.4 Overall Percentage 89.3 Step 2 BUSIER_DOOR Busier door is rear door 100 161 38.3 Busier Door is Front door 41 1566 97.4 Overall Percentage 89.2 Step 3 BUSIER DOOR Busier door is rear door 96 165 36.8 Busier Door is Front door 42 1565 97.4 Overall Percentage 88.9 Step 4 BUSIER DOOR Busier door is rear door 98 163 37.5 Busier Door is Front door 43 1564 97.3 Overall Percentage 89.0 Step 5 BUSIER DOOR Busier door is rear door 95 166 36.4 Busier Door is Front door 40 1567 97.5 Overall Percentage 89.0 a. The cut value is .500 Table B2.6: Omnibus Tests of Model Coefficients of Type B-L.2 Omnibus Tests of Model Coefficients Chi-square df Siq. Step 1 Step 568.401 12 .000 Block 568.401 12 .000 Model 568.401 12 .000 Step 2s Step -.090 1 .765 Block 568.311 11 .000 Model 568.311 11 .000 Step 3s Step -2.070 2 .355 Block 566.241 9 .000 Model 566.241 9 .000 Step & Step -3.314 3 .346 Block 562.928 6 .000 Model 562.928 6 .000 Step 5s Step -.251 1 .617 Block 562.677 5 .000 Model 562.677 5 .000 a- A negative Chi-squares value indicates that the Chi-squares value has decreased from the previous step. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix B3 1. General This appendix illustrate the outputs of the SPSS applied for Multinomial Logistic Regression models proposed in chapter 5, section 5.3.6.2 2. SPPSS Results for Multinomial Logistic Regression Table B3.1: Goodness-of-Fit for Type B-ML.l Goodness-of-Fit Chi-Square df Sig. Pearson 21001.411 12222 .000 Deviance 10360.485 12222 1.000 Table B3.2: Pseudo R-square for Type B-ML.l Pseudo R-Square Cox and Snell .333 Nagelkerke .379 McFadden .193 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 247 Table B3.3: Model Parameters for Type B-ML.l Parameter Estimates 95% Confidence Interval for Exp(B) Busiest door9 B Std. Error Wald df Sig. Exp(B) Lower Bound Upper Bound door 2 busiest Intercept -.133 .123 1.172 1 .279 LOAD_ARR .004 .002 2.911 1 .088 1.004 .999 1.008 LF 0b PUNT .000 .000 .005 1 .945 1.000 1.000 1.000 SEASON -.022 .033 .416 1 .519 .979 .917 1.045 STOP_LOCATE .057 .077 .549 1 .459 1.058 .911 1.230 TIME -.114 .044 6.803 1 .009 .892 .819 .972 TOTAL_ONS -.001 .005 .025 1 .874 .999 .988 1.010 TOTAL_OFFS -.023 .004 36.024 1 .000 .978 .970 .985 Door 1 busiest Intercept 1.302 .121 115.269 1 .000 LOAD_ARR -.006 .002 7.351 1 .007 .994 .990 .998 LF 0b PUNT .000 .000 .502 1 .479 1.000 1.000 1.001 SEASON .067 .032 4.414 1 .036 1.069 1.005 1.139 STOP_LOCATE .549 .075 53.817 1 .000 1.732 1.496 2.006 TIME -.105 .042 6.284 1 .012 .900 .829 .977 TOTAL_ONS .098 .006 269.578 1 .000 1.103 1.090 1.116 TOTAL_OFFS -.308 .010 1013.492 1 .000 .735 .721 .749 a- The reference category is: door 3 busiest. t>. This parameter is set to zero because it is redundant. Table B3.1: Goodness-of-Fit for Type B-ML.2 Goodness-of-Fit Chi-Square df Sig. Pearson 23844.277 10336 .000 Deviance 8352.103 10336 1.000 Table B3.2: Pseudo R-square for Type B-ML.2 Pseudo R-Square Cox and Snell .321 Nagelkerke .372 McFadden .194 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 248 Table B3.3: Model Parameters of Type B-ML.2 Parameter Estimates 95% Confidence Interval for Exp(B) Busiest door8 B Std. Error Wald df Sig. Exp(B) Lower Bound Upper Bound door 2 busiest Intercept -.228 .161 2.020 1 .155 LOAD_ARR .003 .003 1.510 1 .219 1.003 .998 1.008 LF 0b PUNT .000 .000 .003 1 .957 1.000 .999 1.001 SEASON -.011 .039 .082 1 .775 .989 .916 1.067 STOP_LOCATE .034 .090 .139 1 .710 1.034 .867 1.234 TIME -.104 .051 4.142 1 .042 .902 .816 .996 TOTAL_ONS .016 .016 1.004 1 .316 1.016 .985 1.048 TOTAL_OFFS -.023 .006 14.870 1 .000 .978 .967 .989 Total_On_squared .000 .001 .863 1 .353 1.000 .999 1.001 Door 1 busiest Intercept 1.602 .146 121.095 1 .000 LOAD_ARR -.013 .002 30.584 1 .000 .987 .983 .992 LF 0b PUNT .000 .000 .004 1 .947 1.000 .999 1.001 SEASON .122 .035 12.121 1 .000 1.130 1.055 1.210 STOP_LOCATE .659 .084 61.675 1 .000 1.933 1.640 2.279 TIME -.116 .046 6.455 1 .011 .890 .814 .974 TOTAL_ONS .150 .015 95.282 1 .000 1.161 1.127 1.197 TOTAL_OFFS -.300 .010 869.260 1 .000 .741 .726 .756 Total_On_squared -.003 .001 38.672 1 .000 .997 .996 .998 a The reference category is: door 3 busiest. b. This parameter is set to zero because it is redundant. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix C 1. General This Appendix presents the MatLab source codes for the developed model. It includes the source code for bus running time, and boarding and de-boarding passenger predictions. 2. Bus running time function name: running.m function runpre = running(X,Y,Xreal) %Performs bus real-time prediction using modified LOWESS method. %LOWESS was first introduced by Cleveland 1978 and Cleveland et al., 1988 %lnput variables: % % X = data set in form of matrix ( m by d) % each element of this matrix plays running time of the % previous buse recored in the past % there may be d previous buses taken into the pattern and % there are m observations( records) %Xreal= the pattern vector (1 by d) whose each element % is the record of the running time of the previous % bus right before. There are d previous buses. %Y = data set in form of vector (m by 1) whose each element % is the record of running time of the bus in question % in the past.There are m observations. %Outputs: % % runpre = real-time prediction of bus running time, it is a scalar. % %...... % WRITTEN BY: NAM VU, PH.D CANDIDATE, CARLETON UNIVERSITY % START DATE: Jan 15,2006 % FINISH DATE: NO INFORMATION % THIS PROGRAM IS FOR ACADEMIC USES ONLY. %...... % the next lines is to check if the real-time predictions should be made or % not %f=input('Please enter the bus service frequency in second f= '); ______ 249 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 250 % first check fo r the prediction horizon (if d sm aller than 2, there is no % real-time prediction) %rf f >1800 % warning('the bus service frequency is too long'); % fprintf('\nl No real-time estimation being made'); % dum= inputC\nType any key to proceed'); % end % clear f; % NOW HERE IS THE MAIN PROGRAM ...... %load Rpastpattern.dat; % enter history data based on bus ID and service time %load Rpastdata.dat; % enter history running times of the bus %load The Rcurrenpattern.dat % the following lines is to find the optimal bandwidth H by applying % leave-one-out method (see Wofgang 1996) and Nadaraya-Watson estimation % %clear [m,k]=size(X); hmin= round(0.1*m); hmax=round(0.8*m); s= round(0.1*m); for h=hmin:s:hmax for i=1:1:m Y_test=Y(i); Y_train=Y; Y_train(i)=D; test_1=X(i,:); test_2=X; test_2(i,:)=□;% the training set after removing the ith row for n=1 :size(test_2,1) d(n,1)= sqrt(sum((test_2(n,:)-test_1(1,:)).A2)); end disp(' The test set is') disp(Ytrain); disp( 'the Edist to the first second... neighbor') disp(d'); q=sort(d); bandwidth=q(h)% Select the h-nearest neighbours K_weigh=d./bandwidth; for p=1 :length(d) if K_weigh(p,1)<1 w(p,1 )=(70/81 )*(1 -(abs(K_weigh(p,1 ))A3)A3); else w(p,1)=0; end end w; if w==0 warning(’No neighbour has been found'); end y hat =(Y train'*w)/sum(w); ______ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 251 error= (y_hat-Y_test)A2; E(i)=error; end disp('The matrix of Error is') disp(E); Cv =sum(E)/m j=1+(h-hmin)/s; CR0)=Cv; nnear(j)=hmin+(j-1 )*s; end disp(’The CROSS_VALIDATIOn factors') disp(CR) [miCV, pointer]=min(CR); OpNei=rou nd(hm in+s*(poi nter-1)) % ...... PLOTING...... % plot(nnear.CR); xlabel('# of nearest neighbours'),ylabel('CR’); title('h and CR Relationship'); % % PREDICTION STEP...... for i=1 :m eudis0(i,1 )=sqrt(sum((X(i,:)-Xreal(1 ,:)).A2)); end eudisO; B=sort(eudisO); OpBand=B(OpNei); kO=eudisO ./O pBand; fo r i=1 :m if k0(i,1)<1 w0(i,1)=(70/81 )*(1 -(abs(kO(i,1))A3)A3); else w0(i,1)=0; end end wO; BanMa=diag(wO); EstiMa=[ones(m,1 ),X]; IXma=inv(EstiMa'*BanMa*EstiMa) Beta=IXma*EstiMa'*BanMa*Y; slope=Beta; slope(1,:)=D; runpre=Beta(1)+ Xreal*slope %runpre=(Y'*wO)/sum(wO); fprintf(THE PREDCTED BUS RUNNING TIME IS %4.2f SECONDS.’,runpre); %...... % CONFIDENCE INTERVAL ESTIMATION...... de=0; fo r i=1 :m if w0(i,1)~=0 PRE(i,1)=Beta(1)+ (X(i,:) *slope(:,1)); ERRO(i,1 )=(Y(i,1 )-PRE(i,1 ))A2; weigERRO(i, 1 )=ERRO(i, 1 )*w0(i,1); de=de+wO(i,1)A2; end end ______ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 252 de % sum of the squared weights PRE;% Matrix of predicted values of neighbors around xO weigERRO; % Matrix of weighted errors MSE=sum(ERRO) % Matrix of prediction error ns=IXma*EstiMa'*BanMa*BanMa*EstiMa us=EstiMa'*BanMa*BanMa*EstiMa ms=1/(trace(BanMa)-trace(ns)) sigma=sum(weigERRO)*ms bias= (MSE-sigma)/OpNei freDegree=(sum(wO))A2/de fprintf('the degree of freedom is %4.0f .freDegree) critical=input('Please enter the confidence limit critical='); lower=runpre-abs(sqrt(bias))-critical*abs(sqrt(sigma)) upper=runpre-sqrt(bias)+criticarsqrt(sigma) fprintf(THE LOWER LIMIT IS %4.2f seconds at 95 percent confidence level’,lower); fprintf(THE UPPER LIMIT IS %4.2f seconds at 95 percent confidence level’,upper); fprintf(’HAVE A NICE DAY’) 3. Bus boarder function File Name: boarder.m function boarder = nuboarder(X,Y,Xreal) %Performs real-time prediction of number of boarders on the bus using %modified LOW ESS method. %LOWESS was first introduced by Cleveland 1978 and Cleveland et al., 1988 % lnput variables: % % X = Data set in form of matrix ( m by 4) % each element of this matrix plays BOADERS (2 % observations)of the previous buses in same day. % and HEADWAYS (2 observations) of the previous bus and % the bus itself in the past at time k. % There may be m events observed(recorded) % Y = The pattern vector ( 1 by 4) whose elements of this % vector are the records of the boarders in the bus k in % history % and the boarders of the 2 previous buses. % Xreal = Data set in form of vector (m by 1) whose each element % is the predicted runninf time of the bus, the % running time of the bus right before it, the number % of boarders of the 2 previous buses % %Output(s): % % boarder = Predicted number of boarders. It is a scalar. % %...... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 253 % WRITTEN BY NAM VU, PH D CANDIDATE, CARLETON UNIVERSITY % START DATE: MARCH 1,2006 % FINISH DATE: NO INFORMATION % THIS PROGRAM IS FOR ACADEMIC USES ONLY. %...... % the next lines is to check if the real-time predictions should be made or % not f=input('Please enter the bus service frequency in second f= '); % first check for the prediction horizon (if d smaller than 2, there is no % real-time prediction) i f f >1800 warning('the bus service frequency is too long'); fprintfC\nl No real-time estimation being made'); dum= inputC\nType any key to proceed'); end clear f; % ...... NOW HERE IS THE MAIN PRO G RAM ...... %load ('Bpastpattern.dat'); % enter history data based on bus ID and service time %load ('Bpastdata.dat'); % enter history # of boarders of the bus %load (’Bcurrenpattern.dat'); % We should first STANDARDIZE all variables [m,k]=size(X); for j=1 :k % normariize data of past pattern Matrix mX(1 ,j)=mean(X(:,j)); sX(1 ,j)=std(X(:,j)); X(:,j)=(X(:,j)-mX(1,j))/sX(1,j); Xreal(1,j)=(Xreal(1,j)-mX(1,j))/sX(1,j);% Current Pattern is also normalized end %mY=mean(Y); %sY=std(Y); %Y=(Y-mY)/sY; % Exhibit the inputs mX Y disp(Xreal) X hmin= round(0.1*m); hmax=round(0.7*m); s=round(0.1*m); for h=hmin:s:hmax for i=1:1:m Y_test=Y(i); Y_train=Y; Y_train(i)=D; test_1=X(i,:); test_2=X; test_2(i,:)=0;% the training set after removing the ith row ______for n=1:size(test 2,1) ______ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 254 d(n,1)= sqrt(sum((test_2(n,:)-test_1(1,:)).A2)); end disp(' The test set is') disp(Ytrain) disp( 'the Edist to the first second... neighbor') disp(d'); q=sort(d); bandwidth=q(h)% Select the h-nearest neighbours K_weigh=d./bandwidth; for p=1 :length(d) if K_weigh(p,1)<1 w(p,1 )=(70/81 )*(1 -(abs(K_weigh(p, 1 ))A3)A3); else w(p,1)=0; end end w; if w==0 warning('No neighbour has been found'); end y_hat =(Y_train’*w)/sum(w); error= (y_hat-Y_test)A2; E(i)=error; end disp('The matrix of Error is') disp(E); Cv =sum(E)/m j=1 +(h-hmin)/s; CR(j)=Cv; nnear(j)=hmin+(j-1 )*s; end disp('The CROSS_VALIDATIOn factors’) disp(CR) [miCV, pointer]=min(CR); OpNei=round(hmin+s*(pointer-1)) % ...... PLOTING...... % plot(nnear,CR); xlabel('# of nearest neighbours'),ylabel('CR'); title(’h and CR Relationship'); % PREDICTION STEP...... for i=1 :m eudis0(i,1 )=sqrt(sum((X(i,:)-Xreal(1 ,:)).A2)); end eudisO; B=sort(eudisO); OpBand=B(OpNei); kO=eudisO./OpBand; for i=1 :m if k0(i,1)<1 w0(i,1)=(70/81 )*(1-(abs(kO(i,1))A3)A3); else w0(i,1)=0; end end wO;______ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 255 BanMa=diag(wO); EstiMa=[ones(m, 1 ),XJ; IXma=inv(EstiMa'*BanMa*EstiMa); disp(EstiMa'); Beta=IXma*EstiMa'*BanMa*Y; disp(Beta); boarder=round(Y'*wO/sum(wO)) fprintf(2,'THE REAL-TIME PREDCTED # of PASS OF THE BUS IS %4.2f PAS.',boarder); disp('Program complete.HAVE A NICE DAY '); 4. Bus de-boarder function File name: deboarder.m function deboarder = deboarder(X,Y,Xreal) %Performs real-time prediction of number of boarders on the bus using %modified LOWESS method. %LOWESS was first introduced by Cleveland 1978 and Cleveland et al., 1988 %lnput variables: % % X = Data set in form of matrix ( m by 4) % each element of this matrix plays DEBOADERS (2 % observations)of the previous buses in the past. % Y = The pattern vector (1 by 4) whose elements of this % vector are the records of the deboarders in the bus k in % history % and the boarders of the 2 previous buses. % Xreal = Data set in form of vector (m by 1) whose each element % is the the number of deboarder of the 2 previous %buses % %Output(s): % % deboarder = Predicted number of deboarders. It is a scalar. % %...... % WRITTEN BY NAM VU, PH.D CANDIATE, CARLETON UNIVERSITY % START DATE: MARCH 30,2006 % FINISH DATE: NO INFORMATION % THIS PROGRAM IS FOR ACADEMIC USES ONLY. %...... % the next lines is to check if the real-time predictions should be made or % not %f=input('Please enter the bus service frequency in second f= '); % first check for the prediction horizon (if d smaller than 2, there is no % real-time prediction) %if f >1800 % warning('the bus service frequency is too long'); ______ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 256 % fprintfOnl No real-time estimation being made'); % dum= input('\nType any key to proceed'); %end %clear f; % ...... NOW HERE IS THE MAIN PROGRAM ...... %load ('Bpastpattern.dat'); % enter history data based on bus ID and service time %load ('Bpastdata.dat'); % enter history # of boarders of the bus %load ('Bcurrenpattern.dat'); % We should first STANDARDIZE all variables [m,k]=size(X); for j=1:k % normarlize data of past pattern Matrix mX(1 ,j)=mean(X(:j)); sX(1 ,j)=std(X(: j)); X(:,jHX(:,j)-mX(1,j))/sX(1,j); Xreal(1,j)=(Xreal(1,j)-mX(1,j))/sX(1,j);% Current Pattern is also normalized end %mY=mean(Y); %sY=std(Y); %Y=(Y-mY)/sY; % Exhibit the inputs mX Y disp(Xreal) X hmin= round(0.05*m); hmax=round(0.7*m); s= round(0.1*m); for h=hmin:s:hmax for i=1:1:m Y_test=Y(i); Y_train=Y; Y_train(i)=D; test_1=X(i,:); test_2=X; test_2(i,:)=D;% the training set after removing the ith row for n=1 :size(test_2,1) d(n,1 )= sqrt(sum((test_2(n,:)-test_1 (1 ,:)).A2)); end dispf The test set is') disp(Ytrain) disp( 'the Edist to the first second... neighbor') disp(d'); q=sort(d); bandwidth=q(h)% Select the h-nearest neighbours K_weigh=d./bandwidth; for p=1:length(d) if K_weigh(p,1)<1 w(p,1)=(70/81 )*(1-(abs(K_weigh(p,1 ))A3)A3); else w(p,1)=0; end Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 257 end w; if w==0 warning('No neighbour has been found'); end y_hat =(Y_train'*w)/sum(w); error= (y_hat-Y_test)A2; E(i)=error; end disp('The matrix of Error is') disp(E); Cv =sum(E)/m j=1+(h-hmin)/s; CR(j)=Cv; nnear(j)=hmin+(j-1 )*s; end disp('The CROSS_VALIDATIOn factors') disp(CR) [miCV, pointer]=min(CR); OpNei=round(hmin+s*(pointer-1)) % ...... PLOTING...... % plot(nnear,CR); xlabel('# of nearest neighbours'),ylabel('CR'); title('h and CR Relationship'); % PREDICTION STEP...... fo r i=1 :m eudis0(i,1 )=sqrt(sum((X(i,:)-Xreal(1 ,:)).A2)); end eudisO; B=sort(eudisO); OpBand=B(OpNei); kO=eudisO./OpBand; fo r i=1 :m if k0(i,1 )<1 w0(i,1)=(70/81)*(1-(abs(k0(i,1))A3)A3); else w0(i,1)=0; end end wO; BanMa=diag(wO); EstiMa=[ones(m,1 ),X]; IXma=inv(EstiMa'*BanMa*EstiMa); disp(EstiMa'); Beta=IXma*EstiMa'*BanMa*Y; disp(Beta); deboarder=round(Y'*wO/sum(wO)) fprintf(2,'THE REAL-TIME PREDCTED # of PASS OF THE BUS IS %4.2f PAS.',deboarder); disp(’Program complete.HAVE A NICE D AY'); Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix D1 1. General This appendix presents the calibration procedure described in chapter 6. First, two runs were made with VISSIM default parameters (see Table 6.3). By holding the same random seed numbers for each run and the same traffic volume, two other runs were also made with VISSIM but with calibrated values (see Table 6.3). The simulated results on bus travel times then were compared with corresponding actual bus travel times of each bus route provided by the OC Transpo at 95% confidence level. 2. Travel time Measurement Configuration in VISSIM 4.1 Simulated bus travel times were measured by a VISSIM’s tool namely Travel time Measurement Configuration which records travel time of all buses running between stops. A travel time includes actual travel time, stop and go time, dwell time and all delays at intersections. Figure D l.l shows this tool of VISSIM. 258 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 259 TT-Measurement Configuration | X j Active travel tim es: Tune 1 iBilling 5- Holmewoo d Iran: 0 s 2 (Hoimwood-Gtedstoi untl:: [8100' ::S:( 3 f'G!adstone-R'deau.i S fM akezine-K ent) Interval: ] 900; s B iKent-Lebfetonj 7 (Laurerit-Mackezine Agyegation by trrte of | passing the Ostart: . ® destnalion i Output I [2] Compfed i □ R a *1 ; : : i w Table r :Ctrsnspp_TRAVElT!f| OK ] Cancel A Figure D l.l: Travel time measurement tool in VISSIM 3. Calibration Procedure Tables D l.l to D 1.6 present the comparisons between simulated bus travel times and the actual ones provided by the OC Transpo. As shown in the tables, the calibrated parameters provided closer bus running times to the actual data than the use of VISSIM default parameters did. However, if we use the default values suggested by VISSIM, the model still meets calibration targets. Therefore, no further calibration efforts were made after the first adjustment. The parameters used in this study are reasonable. Tables D1.7- D1.10 tabulate VISSIM outputs during calibration process Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 260 4. APC-AVL data from OC Transpo Tables D1.11-D1.16 present actual bus travel time data provided by the OC Transpo Table D l.l: VISSIM Calibration: Billings-IIolmwood Link Simulated data OC transpo data Difference. Significance # of Mean StD # of APC Mean Standard Parameters (1) % Level Simulation (5) (6) records (8) Dev. (9) (2) (3) runs (4) (7) BX ADD=2.0 6.1 0.05 16 464.16 s 31.99 BX MULT=3.0 37 435.6 48 BX ADD=3.0 7.8 0.05 16 469.98 28.38 BX MULT=3.5 Table D1.2: VISSIM Calibration: Holmwood- Gladstone Link Simulated data OC transpo data Significance Diff. % Mean Standard # of APC Mean Standard Parameters (1) Level # of (2) simulation Dev. (6) records Dev. (9) (3) (5) (8) runs (4) (7) BX ADD=2.0 9.49 0.05 16 395.85 36.21 BX MULT=3.0 38 437.4 47.4 s BX ADD=3.0 12.50 0.05 16 382.73 32.68 BX MULT=3.5 Table D1.3: VISSIM Calibration: Gladstone- Rideau Link Simulated data OC transpo data Significance Difference Mean Standard # of APC Mean Standard Parameters (1) Level # of (2) simulation Dev. (6) records Dev. (9) (3) (5) (8) runs (4) (7) BX ADD=2.0 5 0.05 15 595.88 67.52 BX MULT=3.0 35 567.6 s 55.8 s BX ADD=3.0 4.24 0.05 15 591.65 63.37 BX MULT=3.5 Table D1.4: VISSIM Calibration: Laurent- Mackenzie Link Simulated data OC transpo data Significance Difference Parameters (1) Level # of Mean Standard # of APC Mean Standard (2) simulation (5) Dev. records (8) Dev. (9) (3) runs (4) (6) (7) BX ADD=2.0 7.6 0.05 72 611.5 28.7 BX MULT=3.0 104 568.2 s 50.4 BX ADI>=3.0 8.9 0.05 72 618.8 29.1 BX MULT=3.5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 261 Table D1.5: VISSIM Calibration: Mackenzie- Kent Link Simulated data OC transpo data Significance Difference Mean Standa # of APC Mean Standard Parameters (1) Level # of (2) simulation Dev. (9) records Dev. (9)) (3)) (5) (8) runs (4) (6) (7) BX ADD=2.0 10.12 0.05 70 269.1 10.9 BX MULT=3.0 74 299.4 s 30.6 s BX ADD=3.0 15.34 0.05 70 253.5 12.2 BX MULT=3.5 Table D1.6: VISSIM Calibration: Kent- Lebreton Link Simulated data OC transpo data Significance Difference Mean Standard # of APC Mean Standard Parameters (1) Level # of (2) Simulation Dev. records (8) Dev. (9) (3) (5) runs (4) (6) (7) BX ADD=2.0 4.1 0.05 69 153.6 16.1 BX MULT=3.0 74 160.2 19.2 BX ADD=3.0 2.4 0.05 69 156.2 16.7 BX MULT=3.5 Table D1.7: Simulated bus travel time data with VISSIM default values, Random seed =1111 Table o f Travel Times File: c:\myottawa\octranspo2.inp Comment: OC Transpo bus routes 1 and 95 Date: Wednesday, May 10, 2006 11:17:59 AM No. 1 (Bi H i ngs-Hol mewood ): from lin k 344 a t 128.7 m to lin k 127 a t 620.7 m Distance 2119.0 m No. 2 (Hoimwood-Gladstone ): from lin k 127 a t 635.7 m to lin k 10133 a t 4 .7 m Distance 1534.7 m No. 3 (Gladstone-Rideau ): from lin k 116 a t 11.9 m to lin k 417 a t 72.2 m Distance 2148.8 m No. 5 (Makezine-Kent ): from lin k 352 a t 106.1 m to lin k 39 a t 56.3 m Distance 1099.6 m No. 6 (Kent-Lebreton ): from link 39 at 110.2 mto lin k 11 a t 142.5 m Distance 1049.0 m No. 7 (Laurent-Mackezine ): from lin k 350 a t 208.2 m to lin k 20 a t 71.6 m Distance 5227.6 m Time; Trav;#veh; Trav;#Veh; Trav;#Veh; Trav;#Veh; Trav;#Veh; Trav;#Veh; vehc; Rigid bus;; Rigid bus;; Rigid bus;; Articulated bus;; Articulated b us;; Articulated bus;; No.:; 1; 1; 2; 2; 3; 3; 5; 5; 6; 6; 7; 7; Name; B i11i ngs-Holmewood; Bi11ings-Holmewood;Hoimwood-Gl adstone;Hoimwood-Gladstone; Gladstone-Ri deau;Gladstone-Ri deau;Makezi ne-Kent;Makezi ne-Kent;Kent-Lebreton;Kent- Lebreton; Laurent-Mackezi ne;Laurent-Mackezi ne; 900; 0 .0 0; 0 .0 0; 0 .0 0; 254.9; i ; 0.0; 0; 538.8 2 1800; 437.1 1; 0.0 0; 0 .0 0; 224.0; 5; 147.0; 4; 604.8 5 2700; 472.4 1; 404.2 l ; 0 .0 0; 205.1; 4; 159.6; 5; 640.6 5 3600; 469.3 1; 439.3 2; 590.4 2; 220.3; 5; 159.3; 5; 653.2 5 4500; 468.8 2; 344.4 1; 569.2 2; 204.2; 5; 199.7; 5; 625.3 4 5400; 464.8 1; 403.6 2; 737.3 1; 216.7; 4; 139.3; 3; 658.3 4 6300; 476.7 1; 0 .0 0; 629.6 1; 201.7; 3; 150.3; 5; 632.4 4 7200; 462.9 1; 364.4 1; 498.8 l ; 194.4; 4; 152.9; 3; 630.4 3 8100; 0.0 0; 330.3 1; 0 .0 0; 208.7; 4; 137.8; 4; 628.7 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 262 Table D1.8: Simulated bus travel time data with VISSIM defaults values, Random seed =9999 Table o f Travel Times File: c:\myottawa\octranspo2.inp Comment: OC Transpo bus routes 1 and 95 Date: Wednesday, May 10, 2006 12:10:47 PM No. 1 (Billings-Holmewood ) : from lin k 344 at 128.7 m to link 127 at 620.7 m, Distance 2119.0 m No. 2 (Hoimwood-Gladstone ) : from lin k 127 at 635.7 m to link 10133 at 4.7 m, Distance 1534.7 m No. 3 (Gladstone-Rideau ): from link 116 at 11.9 m to link 417 at 72.2 m, Distance 2148.8 m No. 5 (Makezine-Kent ): from link 352 a t 106.1 m to lin k 39 a t 56.3 m, Distance 1099.6 m No. 6 (Kent-Lebreton ) : from lin k 39 a t 110.2 m to lin k 11 a t 142.5 m, Distance 1049.0 m No. 7 (Laurent-Mackezine ): from link 350 a t 208.2 m to lin k 20 at 71.6 m, Distance 5227.6 m Time; Trav;#Veh; Trav;#veh; Trav;#Veh; Trav;#veh; Trav;#Veh; Trav;#veh; vehc; Rigid bus;; Rigid bus;; Rigid bus;; Articulated bus;; Articulated bus;; Articulated bus;; No.:; 1; 1; 2; 2; 3; 3; 5; 5; 6; 6; 7; 7; Name;B i11i ngs-Holmewood; B i11i ngs-Holmewood;Hoimwood-Gladstone;Hoimwood- Gl adstone;Gladstone-Ri deau;Gladstone-Ri deau;Makezi ne-Kent;Makezi ne-Kent;Kent 900; 0.0 0; 0 .0 0; 0 .0 0 187.9 1; 0.0 0; 542.7 2 1800; 0.0 0; 0 .0 0; 0 .0 0 225.6 4; 166.9 5; 588.7 5 2700; 477.2 2; 401.3 1; 0.0 0 228.2 5; 154.6 4; 593.4 5 3600; 467.9 2; 369.9 2; 529.0 2 201.4 6; 152.7 5; 628.4 5 4500; 565.1 1; 391.9 1; 551.0 1 195.5 4; 163.3 5; 618.0 4 5400; 455.1 1; 369.2 2; 625.7 2 202.9 4; 156.2 4; 628.1 3 6300; 435.2 1; 0.0 0; 555.5 1 217.9 3; 174.1 3; 652.6 5 7200; 453.3 1; 391.0 1; 694.9 1 201.4 4; 133.8 5; 588.2 3 8100; 0.0 0; 332.3 1; 579.0 1 209.3 4; 132.4 3; 598.6 4 Table D1.9: Simulated bus travel time data with VISSIM Calibrated values, Random seed =1111 Table of Travel Times Fi1e: c:\myottawa\octranspo2.i np Comment: OC Transpo bus routes 1 and 95 Date: Wednesday, May 10, 2006 3:05:00 PM No. 1 (Billings-Holmewood ): from lin k 344 a t 128.7 mto lin k 127 a t 620.7 m Distance 2119.0 m No. 2 (Hoimwood-Gladstone ): from lin k 127 a t 635.7 m to lin k 10133 a t 4 .7 m Distance 1534.7 m no. 3 (Gladstone-Rideau ): from lin k 116 a t 11.9 m to lin k 417 a t 72.2 m Distance 2148.8 m No. 5 (Makezine-Kent ): from lin k 352 a t 106.1 m to lin k 39 a t 56.3 m Distance 1099.6 m No. 6 (Kent-Lebreton ): from lin k 39 a t 110.2 m to lin k 11 a t 142.5 m Distance 1049.0 m No. 7 (Laurent-Mackezine ): from lin k 350 a t 208.2 mto lin k 20 a t 71.6 m Distance 5227.6 m Time; Trav;#Veh; Trav;#veh; Trav;#veh; Trav;#Veh; Trav;#veh; Trav;#veh; Vehc; Rigid bus;; Rigid bus;; Rigid bus;; Articulated bus;; Articulated bus;; Articulated bus;; No.:; 1; 1; 2; 2; 3; 3; 5; 5; 6; 6; 7; 7; Name;B i11i ngs-Holmewood; Bi11ings-Holmewood;Hoimwood-Gladstone;Hoimwood- Gl adstone;Gladstone-Ri deau;Gladstone-Ri deau;Makezi ne-Kent;Makezi ne-Kent;Kent- Lebreton ;Kent-Lebreton;Laurent-Mackezine;Laurent-Mackezi ne; 900; 0.0 0; 0.0 0; 0.0 0; 232.4 l ; 0 .0 0; 539.9 3 1800; 434.5 1; 0.0 0; 0.0 0; 221.5 5; 146.3 4; 600.7 4 2700; 471.9 1; 414.9 1; 0 .0 0; 209.6 4; 156.6 5; 640.6 5 3600; 468.2 2; 440.1 2; 621.6 2; 208.4 6; 153.8 5; 589.8 5 4500; 460.4 1; 354.1 1; 566.4 2; 209.3 4; 196.7 5; 643.7 4 5400; 462.8 1; 442.3 2; 667.0 1; 202.9 4; 142.6 4; 634.3 4 6300; 468.2 1; 0.0 0; 631.2 1; 213.8 3; 139.3 4; 604.1 4 7200; 464.9 1; 374.8 1; 496.7 1; 188.5 4; 170.5 3; 632.2 3 8100; 0.0 0; 434.3 1; 0.0 0; 203.7 4; 131.9 4 ; 628.7 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 263 Table D1.10: Simulated Bus Travel Time data with VISSIM Calibrated Values, Random seed =9999 Table o f Travel Times Fi1e: c:\myottawa\octranspo2.i np Comment: OC Transpo bus routes 1 and 95 Date: Wednesday, May 10, 2006 2:16:14 PM NO. 1 (B i11i ngs-Holmewood ): from 1ink 344 at 128.7 mto link 127 a t 620.7 m Distance 2119.0 m NO. 2 (Hoimwood-Gladstone ): from 1ink 127 a t 635.7 mto link 10133 a t 4 .7 m Distance 1534.7 m No. 3 (Gladstone-Ri deau ): from 1ink 116 at 11.9 m to lin k 417 a t 72.2 m Distance 2148.8 m No. 5 (Makezine-Kent ): from 1ink 352 a t 106.1 mto link 39 a t 56.3 m Di stance 1099.6 m No. 6 (Kent-Lebreton ): from 1ink 39 a t 110.2 m to lin k 11 a t 142.5 m Di stance 1049.0 m No. 7 (Laurent-Mackezi ne ): from lin k 350 a t 208.2 m to lin k 20 a t 71.6 m Distance 5227.6 m Time; T rav;#veh; Trav;#veh; Trav;#Veh; Trav;#Veh; Trav;#veh; Trav;#veh; Vehc; Rigid bus;; Rigid bus;; Rigid bus;; Articulated bus;; Articulated bus;; Articulated bus;; No.:; 1; 1; 2; 2; 3; 3; 5; 5; 6; 6; 7; 7; Name;Bi11ings-Holmewood;Bi11i ngs-Holmewood;Hoimwood-Gladstone;Hoimwood- Gl adstone;Gladstone-Ri deau;Gladstone-Ri deau;Makezi ne-Kent;Makezi ne-Kent;Kent- Lebreton ;Kent-Lebreton;Laurent-Mackezi ne;Laurent-Mackezi ne; 900; 0.0 0; 0.0 0; 0.0 0; 191.3 1; 0.0 0; 542.5 2 1800; 0.0 0; 0.0 0; 0.0 0; 225.9 4; 158.3 5; 588.5 5 2700; 477.3 2; 367.1 1; 0.0 0; 228.2 5; 144.7 4; 593.4 5 3600; 455.3 2; 374.9 2; 530.3 2; 201.5 6; 146.4 5; 628.8 5 4500; 564.0 1; 377.6 1; 549.9 1; 194.7 4; 159.8 5; 617.7 4 5400; 410.2 1; 391.3 2; 621.4 2; 203.1 4; 156.4 4; 628.1 3 6300; 435.3 1; 0 .0 0; 554.1 1; 217.1 3; 170.2 3; 652.6 5 7200; 452.8 1; 390.7 1; 771.4 1; 201.4 4; 150.2 5; 588.2 3 8100; 0 .0 0; 323.0 1; 588.5 1; 209.3 4; 130.0 4; 598.6 4 Table DI. 11; Actual Bus travel time: Billings- Holmwood stops PROCESSED: 2 0 0 6 -0 5 -0 9 1 3 :1 1 :3 5 AUTOMATIC PASSENGER COUNTING SYSTEM REVENUE TIME UTILIZATION WEEKDAY SERVICE BOOKING: APR05 FROM STOP RA945 BILLINGS BRIDGE 4C PERIOD: 2005-04-24 TO 2005-06- 25 TO STOP: CF630 BANK HOLMWOOD FS DAY TYPE: WEEKDAY ROUTE: 1 Bank - - Rockcliffe TIME: 08:00: 00 TO 1 0 :0 0 :0 0 A v erag e Time p e r T r ip S td % o f M in u te s Dev Tim e Moving between stops 4 .85 0 .8 6 6 6 .7 3 Stop and go time 0 .0 5 0 .2 6 0 .7 4 I d l e tim e 0 .4 8 0 .6 7 6 .5 6 D w ell tim e 0 .8 3 0 .3 5 1 1 .4 0 Excess time 1 .0 6 0 .4 9 1 4 .5 6 TOTAL 7 .2 6 0 .8 0 100% Layover time 2 .5 8 Average sched time per trip 5 .3 2 Total distance (KM) 2 .1 2 A v e rag e m oving s p e e d (KM/HR) 2 6 .2 4 A v e rag e t o t a l s p e e d (KM/HR) 1 7 .5 1 Total trips captured: 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 264 Table DI. 12: Actual Bus travel time: Holmwood- Gladstone stops PROCESSED 2 0 0 6 -0 5 -0 9 1 3 :1 1 :5 5 AUTOMATIC PASSENGER COUNTING SYSTEM REVENUE TIME UTILIZATION WEEKDAY SERVICE BOOKING: APR05 FROM STOP: CF630 BANK HOLMWOOD FS PERIOD: 2005-04-24 TO 2005-06-25 TO STOP: CH080 BANK GLADSTONE NS DAY TYPE: WEEKDAY ROUTE: 1 Bank - - Rockcliffe IME: 08:00:00 TO 10:00:00 A v erag e Tim e p e r T r ip S td % o f M in u te s Dev Time Moving between stops 3 .9 0 1.13 53.57 Stop and go time 0 .2 8 0 .6 1 3 .7 9 I d l e tim e 0 .0 4 0 .1 9 0 .6 0 Dwell time 1 .2 7 0.38 17.46 Excess time 1 .7 9 0 .5 3 2 4 .5 7 TOTAL 7 .2 9 0 .7 9 100% Layover time 0 .0 0 Average sched time per trip 5 .0 0 Total distance (KM) 1 .5 5 A v e ra g e m oving s p e e d (KM/HR) 2 3 .8 3 A v e ra g e t o t a l s p e e d (KM/HR) 1 2 .7 7 Total trips captured: 38 Table DI. 13: Actual Bus travel time: Gladstone-Rideau PROCESSED 2 0 0 6 -0 5 -0 9 1 3 :1 2 :1 6 AUTOMATIC PASSENGER COUNTING SYSTEM REVENUE TIME UTILIZATION WEEKDAY SERVICE BOOKING: APR05 FROM STOP: CH080 BANK GLADSTONE NS PERIOD: 2005-04-24 TO 2005-06-25 TO STOP: CD920 RIDEAU 3A DAY TYPE: WEEKDAY ROUTE: 1 Bank - - R o c k c lif f e TIME: 08:00:00 TO 10:00:00 A v e rag e Tim e p e r T r ip S td % o f M in u te s Dev Time Moving between stops 6 .4 3 2.05 67.93 Stop and go time 0 .5 8 1 .4 6 6 .1 2 I d l e tim e 0 .3 0 0 .5 4 3 .1 2 Dwell time 1 .0 5 0.32 11.05 Excess time 1 .1 1 0 .6 0 1 1 .7 8 TOTAL 9 .4 6 0 .9 3 100% Layover time 0 .0 0 Average sched time per trip 1 3 .0 0 Total distance (KM) 2 .1 0 A v e ra g e m oving s p e e d (KM/HR) 1 9 .6 0 A v e ra g e t o t a l s p e e d (KM/HR) 1 3 .3 2 Total trips captured: 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 265 Table DI. 14: Actual Bus travel time: St.Laurent-Mackenzie stops PROCESSED: 2 0 0 6 -0 5 -0 9 1 3 :1 1 :1 1 AUTOMATIC PASSENGER COUNTING SYSTEM REVENUE TIME UTILIZATION WEEKDAY SERVICE BOOKING: APR05 FROM STOP: EB905 ST LAURENT 2B PERIOD: 2005-04-24 TO 2 0 0 5 -0 6 -2 5 TO STOP: CD910 MACKENZIE KING 2A DAY TYPE: WEEKDAY ROUTE: 95 T r a n s itw a y TIME: 08:00:00 TO 10: 0 0 :0 0 A v erag e Time p e r T r ip S td % o f M in u te s Dev Time Moving between stops 8.17 0.75 86.29 Stop and go time 0.03 0.19 0.33 I d l e tim e 0 .1 3 0 .3 2 1 .3 5 D w ell tim e 0 .6 9 0 .3 4 7 .2 4 Excess time 0 .4 5 0 .1 7 4 .7 9 TOTAL 9 .4 7 0 .8 4 100% Layover time 0 .0 0 Average sched time per trip 8 .6 9 Total distance (KM) 5 .2 2 A v e rag e m o v in g s p e e d (KM/HR) 3 8 .3 2 A v e rag e t o t a l s p e e d (KM/HR) 3 3 .0 7 Total trips captured: 104 Table DI. 15: Actual Bus travel time: Mackenzie- Kent stops PROCESSED: 2 0 0 6 -0 5 -0 9 1 3 :1 1 :1 0 AUTOMATIC PASSENGER COUNTING SYSTEM REVENUE TIME UTILIZATION WEEKDAY SERVICE BOOKING: APR05 FROM STOP: CD910 MACKENZIE KING 2A PERIOD: 2005-04-24 TO 2005-06-25 TO STOP: CA920 ALBERT KENT DAY TYPE: WEEKDAY ROUTE: 95 T r a n s itw a y TIME: 08:00:00 TO 1 0 :0 0 :0 0 A v erag e Time p e r T r ip S td % o f Minutes Dev Time ------ Moving between stops 3.55 0.53 7 1 .0 9 Stop and go time 0 .0 1 0 .0 6 0 .1 4 I d l e tim e 0 .1 9 0 .3 9 3 .8 6 D w ell tim e 0.64 0.31 12.75 Excess time 0 .6 1 0 .2 9 1 2 .1 6 TOTAL 4.99 0.51 100% Layover time 0.00 Average sched time per trip 4.72 Total distance (KM) 1 .1 9 A v e rag e m oving s p e e d (KM/HR) 2 0 .1 2 A v e rag e t o t a l s p e e d (KM/HR) 1 4 .3 0 Total trips captured: 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 266 Table DI. 16: Actual Bus travel time: Kent - Lebreton stops PROCESSED: 2 0 0 6 -0 5 -1 1 1 1 :4 0 :5 7 AUTOMATIC PASSENGER COUNTING SYSTEM REVENUE TIME UTILIZATION WEEKDAY SERVICE BOOKING: APR05 FROM STOP: CA920 ALBERT KENT PERIOD : 2005-04-24 TO 2005-06-25 TO STOP: C J900 LEBRETON 1A DAY TYPE: WEEKDAY ROUTE: 95 T r a n s itw a y TIME: 08:00:00 TO 10:00:00 A v erag e Time p e r T r ip S td % o f Minutes Dev Time Moving between stops 2 .4 5 0 .3 6 9 1 .5 6 Stop and go time 0.03 0.19 1.14 I d l e tim e 0 .0 0 0 .0 0 0 .0 0 D w ell tim e 0 .0 9 0 .0 8 3 .3 7 Excess time 0.10 0.08 3.93 TOTAL 2 .6 7 0 .3 2 100% Layover time 0 .0 0 Average sched time per trip 5 .0 0 Total distance (KM) 1 .0 6 A v e rag e m o v in g s p e e d (KM/HR) 2 6 .0 1 A v e rag e t o t a l s p e e d (KM/HR) 2 3 .8 2 Total trips captured: 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix D2 1. General This Appendix presents the VISSIM’s outputs following simulation and processing. There were 170 tables for the two bus routes. These record the operation information of the buses. Because the data are large1, only samples are printed out for the sake of illustration. 2. Legend t= Arrival Time at stop (second) ToD= Time of Day; STime =Start Time of the course (trip) at the first stop (second) Line= Bus line Stp Stop Number StpBd=Number of boarding Passengers StpAlt= Number of Alighting Passengers StpDwl= Stop Dwell Time (second) StpSvcT= Stop Service Time (second) StpWP Waiting Passenger Bus Station Name Code (Stp) Line St Laurent 2A 51 95 Train 2A 16 95 Hurdman 20 95 Lees 21 95 Campus 50 95 Laurier 25 95 Mackenzie King 23 95 Billings Bridge 1 1 Bank-Holmwood 54 1 Bank-Gladstone 3 1 Rideau Center 53 1 1 It could be about 1200 pages if all of the above tables are printed out. 267 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 268 Table D2.1: Route 95- Situation 1200 26 07:45:26.0 0.5 95 1 ! 51 0 8 27 27 126.5 07:47:06.5 0.5 95 1 ! 16 0 8 15.4080185674615 15.4080185674615 206 07:48:26.0 180.5 95 2 I 51 10 8 52 52 229 : 07:48:49.0 0.5 95 1 | 20 14 9 64.5 64.5 331.5 i 07:50:31.5 180.5 95 2 16 10 8 16.6408717933666 16.6408717933666 355.5 i 07:50:55.5 0.5 95 1 I 21 14 9 20.7682394096475 20.7682394096475 386.5 ; 07:51:26.5 360.5 95 3 51 11 8 54.5 54.5 450.5 07:52:30.5 0.5 95 1 50 14 9 41.1177085856324 41.1177085856324 437.5 07:52:17.5 180.5 95 2 20 12 12 67 67 521.5 i 07:53:41.5 0.5 95 1 25 14 9 11.8510497799656 11.8510497799656 568.5 07:54:28.5 180.5 95 2 21 12 12 17.6706932648818 17.6706932648818 516 07:53:36.0 360.5 95 3 16 11 8 13.0275490869938 13.0275490869938 566 i 07:54:26.0 540.5 95 4 51 7 8 44.5 44.5 619.5 07:55:19.5 360.5 95 3 20 10 12 62 62 652 07:55:52.0 0.5 95 1 23 18 18 97 97 661 ' 07:56:01.0 180.5 95 2 50 12 12 20.1698061319809 20.1698061319809 711.5 : 07:56:51.5 180.5 95 2 25 12 12 9.41338555159671 9.41338555159671 683 07:56:23.0 540.5 95 4 16 7 8 20.6037198994942 20.6037198994942 734.5 i 07:57:14.5 360.5 95 3 21 0 0 0 0 747 ! 07:57:27.0 720.5 95 5 51 6 8 42 42 792.5 07:58:12.5 540.5 95 4 20 11 11 62 62 809.5 . 07:58:29.5 360.5 95 3 50 0 0 19.1968338518466 19.1968338518466 820 ; 07:58:40.0 0.5 95 1 42 18 18 29.2409836316985 29.2409836316985 773 i 07:57:53.0 i 180.5 95 2 23 4 21 69.5 69.5 865.5 07:59:25.5 I 720.5 95 5 16 6 8 30.6395278375851 30.6395278375851 883.5 : 07:59:43.5 0.5 95 1 24 18 18 32.0886676006806 32.0886676006806 859.5 ! 07:59:19.5 I 360.5 95 3 25 0 0 25.8094241290089 25.8094241290089 918 08:00:18.0 I 540.5 95 4 21 11 11 23.7811306750596 23.7811306750596 926 08:00:26.0 I 900.5 95 6 51 4... 8 37 37 941 08:00:41.0 I 180.5 95 2 42 4 21 9.55396359766676 9.55396359766676 953 08:00:53.0 0.5 95 1 7 13 25 102 102 989 08:01:29.0 i 720.5 95 5 20 18 11 79.5 79.5 993.5 ! 08:01:33.5 I 180.5 95 2 24 4 21 26.2339346047374 26.2339346047374 1013.5 I 08:01:53.5 ! 360.5 95 3 23 6 20 72 72 1016.5 I 08:01:56.5 ! 540.5 95 4 50 11 11 25.8639488739813 25.8639488739813 1038 ( 08:02:18.0 i 900.5 95 6 16 4 8 19.4601644732678 19.4601644732678 1072 08:02:52.0 I 540.5 95 4 25 11 11 29.635358426345 29.635358426345 1074 08:02:54.0 I 180.5 95 2 7 2 17 54.5 54.5 1085.5 08:03:05.5 0.5 95 I 1 15 13 25 34.943301091224 34.943301091224 1105 08:03:25.0 I 1080.5 95 7 51 19 ! 8 74.5 74.5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 269 1135 08:03:55.0 720.5 95 5 21 18 11 14.6498056895333 14.6498056895333 1141 08:04:01.0 540.5 95 4 23 6 19 69.5 69.5 1148.5 08:04:08.5 900.5 95 6 20 8 10 52 52 1182.5 08:04:42.5 180.5 95 2 15 2 17 29.3198984205231 29.3198984205231 1180.5 08:04:40.5 360.5 95 3 42 6 20 7.28735577834652 7.28735577834652 1234 08:05:34.0 360.5 95 3 24 6 20 11.268640117897 11.268640117897 1251 08:05:51.0 1080.5 95 7 16 19 8 9.91379574376476 9.91379574376476 1258 08:05:58.0 0.5 95 1 52 13 25 7.94756682398519 7.94756682398519 1228.5 08:05:28.5 720.5 95 5 50 18 11 19.6868938284895 19.6868938284895 1266.5 08:06:06.5 900.5 95 6 21 8 10 11.1554981443269 11.1554981443269 1279.5 08:06:19.5 720.5 95 5 25 18 11 40.5561034458096 40.5561034458096 1286.5 08:06:26.5 1260.5 95 8 51 12 8 57 57 1300.5 08:06:40.5 540.5 95 4 42 6 19 35.219331599476 35.219331599476 1314 08:06:54.0 360.5 95 3 7 2 18 57 57 1319 08:06:59.0 180.5 95 2 52 2 17 33.9811784389407 33.9811784389407 1349.5 08:07:29.5 1080.5 95 7 20 6 15 59.5 59.5 1354 08:07:34.0 900.5 95 6 50 8 10 25.3888255677701 25.3888255677701 1369 08:07:49.0 540.5 95 4 24 6 19 5.49294724121229 5.49294724121229 1375.5 08:07:55.5 720.5 95 5 23 5 22 74.5 74.5 1410.5 08:08:30.5 900.5 95 6 25 8 10 21.915413922672 21.915413922672 1417 08:08:37.0 1260.5 95 8 16 12 8 19.9467728345976 19.9467728345976 1422.5 08:08:42.5 360.5 95 3 15 2 18 13.9579757504817 13.9579757504817 1433.5 08:08:53.5 540.5 95 4 7 3 18 59.5 59.5 1472 08:09:32.0 1080.5 95 7 21 6 15 22.2812288908245 22.2812288908245 1493.5 08:09:53.5 900.5 95 6 23 5 17 62 62 1467 08:09:27.0 1440.5 95 9 51 8 8 47 47 1525.5 08:10:25.5 1260.5 95 8 20 5 13 52 52 1542 08:10:42.0 720.5 95 5 42 5 22 27.7101882253563 27.7101882253563 1542 08:10:42.0 540.5 95 4 15 3 18 14.9754120586199 14.9754120586199 1559 08:10:59.0 360.5 95 3 52 2 18 18.2594138743781 18.2594138743781 1591.5 08:11:31.5 1440.5 95 9 16 8 8 21.7979516728509 21.7979516728509 1566.5 08:11:06.5 1080.5 95 7 50 6 15 28.223327691404 28.223327691404 1605.5 08:11:45.5 720.5 95 5 24 5 22 2.03845422638412 2.03845422638412 1624 08:12:04.0 1080.5 95 7 25 6 15 29.4021079828607 29.4021079828607 1641 08:12:21.0 1260.5 95 8 21 5 13 11.4381248583569 11.4381248583569 1646 08:12:26.0 1620.5 95 10 51 16 8 67 67 1661 08:12:41.0 900.5 95 6 42 5 17 21.4709414145211 21.4709414145211 1674 08:12:54.0 720.5 95 5 7 2 19 59.5 59.5 1678.5 08:12:58.5 540.5 95 4 52 3 18 24.2338642213828 24.2338642213828 1716.5 I 08:13:36.5 900.5 95 6 24 5 17 28.1892212176801 28.1892212176801 1707.5 08:13:27.5 1440.5 95 9 20 10 12 62 62 1727 08:13:47.0 1260.5 95 8 50 5 13 11.8356045866382 11.8356045866382 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 270 1731.5 08:13:51.5 1080.5 95 7 23 7 21 77 77 1768.5 o 00 ro 00 in 1260.5 95 8 25 5 13 52.8966901428382 52.8966901428382 1783.5 08:14:43.5 720.5 95 5 15 2 19 25.2340207650103 25.2340207650103 1785.5 08:14:45.5 1620.5 95 10 16 16 8 22.960157967385 22.960157967385 1826 08:15:26.0 1800.5 95 11 51 15 8 64.5 64.5 1794.5 08:14:54.5 900.5 95 6 7 0 15 44.5 44.5 1837 08:15:37.0 1440.5 95 9 21 10 12 52.824036320219 52.824036320219 1861 08:16:01.0 1260.5 95 8 23 4 18 62 62 1897 08:16:37.0 1620.5 95 10 20 8 14 62 62 1901.5 08:16:41.5 1080.5 95 7 42 7 21 13.7412054252605 13.7412054252605 1906 08:16:46.0 900.5 95 6 15 0 15 24.8785079302955 24.8785079302955 1920.5 08:17:00.5 720.5 95 5 52 2 19 0 0 1952.5 08:17:32.5 1080.5 95 7 24 7 21 29.4062008271227 29.4062008271227 1965 08:17:45.0 1800.5 95 11 16 15 8 3.4983609929123 3.4983609929123 1969 08:17:49.0 1440.5 95 9 50 10 12 1.6792013983905 1.6792013983905 2003 08:18:23.0 1440.5 95 9 25 10 12 20.219276874692 20.219276874692 2019 08:18:39.0 1260.5 95 8 42 4 18 25.3860101450702 25.3860101450702 2022 08:18:42.0 1620.5 95 10 21 8 14 17.1548377015565 17.1548377015565 2006.5 08:18:26.5 1980 i 95 12 51 24 8 87 87 2034.5 08:18:54.5 iu8u. j 95 7 7 7 19 72 72 2039.5 08:18:59.5 900.5 95 6 52 0 15 19.2632181295951 19.2632181295951 2077.5 08:19:37.5 1260.5 95 8 24 4 18 21.8194859816282 21.8194859816282 2059 08:19:19.0 1800.5 95 11 20 8 14 62 62 2095.5 08:19:55.5 1440.5 95 9 23 14 19 89.5 89.5 2113 08:20:13.0 1620.5 95 10 50 8 14 20.908244199681 20.908244199681 2138.5 08:20:38.5 1080.5 95 7 15 7 19 14.7975736539457 14.7975736539457 2153.5 08:20:53.5 1260.5 95 8 7 3 15 52 52 2163 08:21:03.0 1620.5 95 10 25 8 14 35.3233184129663 35.3233184129663 2167.5 08:21:07.5 1980.5 95 12 24 8 20.5656558967728 20.5656558967728 2185 08:21:25.0 2160.5 95 13 11 8 54.5 54.5 2186 08:21:26.0 1800.5 95 11 8 14 17.2122399097436 17.2122399097436 2235.5 08:22:15.5 1080.5 95 7 7 19 34.7580535812084 34.7580535812084 2263.5 08:22:43.5 1440.5 95 9 14 19 19.1026094806307 19.1026094806307 2265 08:22:45.0 1260.5 95 8 ! 3 15 27.0040373374394 27.0040373374394 2278 08:22:58.0 1980.5 95 12 ! 9 16 69.5 69.5 2279.5 08:22:59.5 1800.5 95 11 8 14 26.5560844710106 26.5560844710106 2319 08:23:39.0 1440.5 95 9 14 19 22.707752223182 22.707752223182 2312 08:23:32.0 2160.5 95 13 11 8 25.1572047241585 25.1572047241585 2332 08:23:52.0 1620.5 95 10 11 21 87 87 2337 08:23:57.0 1800.5 95 11 8 14 18.3323028581116 18.3323028581116 2366.5 08 !4:26.5 2340.5 95 14 51 22 8 82 82 2395 08 14:55.0 1440.5 95 9 7 4 23 74.5 74.5 2398.5 08:24:58.5 1260.5 95 8 52 3 15 11.7219697401678 11.7219697401678 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 271 2425 08:25:25.0 2160.5 95 13 20 7 12 54.5 54.5 2412 08:25:12.0 1980.5 95 12 21 9 16 24.2663476350826 24.2663476350826 2453.5 08:25:53.5 1800.5 95 11 23 8 20 77 77 2502.5 08:26:42.5 1440.5 95 9 15 4 23 35.7978008883888 35.7978008883888 2512.5 08:26:52.5 1980.5 95 12 50 9 16 34.9375145429651 34.9375145429651 2499.5 08:26:39.5 1620.5 95 10 42 11 21 6.28853073269999 6.28853073269999 2521 08:27:01.0 2340.5 95 14 16 22 8 20.2395481154767 20.2395481154767 2545.5 08:27:25.5 2520.5 95 15 51 21 8 79.5 79.5 2542.5 08:27:22.5 2160.5 95 13 21 7 12 27.474141306356 27.474141306356 2552.5 08:27:32.5 1620.5 95 10 24 11 21 20.4017029958996 20.4017029958996 2577.5 08:27:57.5 1980.5 95 12 25 9 16 23.2010066579892 23.2010066579892 2621 08:28:41.0 1800.5 95 11 42 8 20 16.2928305636922 16.2928305636922 2628 08:28:48.0 2340.5 95 14 20 12 16 77 77 2634 08:28:54.0 1620.5 95 10 7 5 22 74.5 74.5 2642 08:29:02.0 2160.5 95 13 50 7 12 4.9348794933683 4.9348794933683 2674 08:29:34.0 1800.5 95 11 24 8 20 22.9099954820845 22.9099954820845 2676 08:29:36.0 2160.5 95 13 25 7 12 23.468804485881 23.468804485881 2693 08:29:53.0 1980.5 95 12 23 16 24 107 107 2698 08:29:58.0 2520.5 95 15 16 21 8 22.6049955504088 22.6049955504088 2700.5 08:30:00.5 1440.5 95 9 52 4 23 0 0 2726.5 08:30:26.5 2700.5 95 16 51 18 8 72 72 2740 08:30:40.0 1620.5 95 10 15 5 22 21.1668147437831 21.1668147437831 2767 08:31:07.0 2340.5 95 14 21 12 16 36.6406023740624 36.6406023740624 2754 08:30:54.0 1800.5 95 11 7 2 20 62 62 2812.5 08:31:52.5 2160.5 95 13 23 1 19 57 57 2809 08:31:49.0 2520.5 95 15 20 13 15 77 77 2873 08:32:53.0 2700.5 95 16 16 18 8 25.8976918188092 25.8976918188092 2878 08:32:58.0 1620.5 95 10 52 5 22 32.1622888171239 32.1622888171239 2878 08:32:58.0 2340.5 95 14 50 12 16 5.67213631849137 5.67213631849137 2862.5 08:32:42.5 1980.5 95 12 42 16 24 9.55724702847379 9.55724702847379 2867 5 08 3 ?:47.5 1800.5 95 11 15 2 20 6.78266611184534 6.78266611184534 290< 5 08:33:26.5 2880.5 95 17 51 21 8 79.5 79.5 2913 u8.o3:33.0 2340.5 95 14 25 12 16 25.7887515919864 25.7887515919864 2913.5 08:33:33.5 1980.5 95 12 24 16 24 7.00857075452838 7.00857075452838 2949 08:34:09.0 2520.5 95 15 21 13 15 22.5148356183951 22.5148356183951 2957.5 08:34:17.5 1800.5 95 11 52 2 20 17.7503283888678 17.7503283888678 2980.5 08:34:40.5 2160.5 95 13 42 1 19 39.1189068557031 39.1189068557031 2989 5 08:34:49.5 2700.5 95 16 20 13 15 77 77 2993 5 08:34:53.5 1980.5 95 12 7 3 28 84.5 84.5 3045.5 08:35:45.5 2520.5 95 15 50 13 15 26.1703266971289 26.1703266971289 3060.5 08:36:00.5 2880.5 95 17 16 21 8 12.9772107090858 12.9772107090858 3052.5 08:35:52.5 2340.5 95 14 23 7 25 87 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 272 3086 08:36:26.0 3060.5 95 18 51 28 8 97 97 3101.5 08:36:41.5 2520.5 95 15 25 13 15 24.9566980658624 24.9566980658624 3092.5 08:36:32.5 2160.5 95 13 24 1 19 19.5688205404824 19.5688205404824 3119 08:36:59.0 1980.5 95 12 15 3 28 19.2069067504596 19.2069067504596 3131.5 08:37:11.5 2700.5 95 16 21 13 15 28.5398107125153 28.5398107125153 3163.5 08:37:43.5 2880.5 95 17 20 9 15 67 67 3172.5 08:37:52.5 2520.5 95 15 23 3 25 77 77 3173.5 08:37:53.5 2160.5 95 13 7 1 14 44.5 44.5 3218.5 08:38:38.5 2340.5 95 14 42 7 25 43.3106773439273 43.3106773439273 3239 08:38:59.0 1980.5 95 12 52 3 28 7.28084833846673 7.28084833846673 3236.5 08:38:56.5 2700.5 95 16 50 13 15 33.0503709327271 33.0503709327271 3267 08:39:27.0 3240.5 95 19 51 19 9 77 77 3256.5 08:39:16.5 3060.5 95 18 16 28 8 11.7016795737678 11.7016795737678 3283.5 08:39:43.5 2160.5 95 13 15 1 14 38.4944297014503 38.4944297014503 3294.5 08:39:54.5 2880.5 95 17 21 9 15 45.2037699409313 45.2037699409313 3300.5 08:40:00.5 2700.5 95 16 25 13 15 26.6685695871849 26.6685695871849 3339 08:40:39.0 2520.5 95 15 42 3 25 0.143383196888053 0.143383196888053 3332.5 08:40:32.5 2340.5 95 14 24 7 25 18.2476665302477 18.2476665302477 3357 08:40:57.0 3060.5 95 18 20 5 18 64.5 64.5 3393 08:41:33.0 2520.5 95 15 24 3 25 17.5281466034446 17.5281466034446 3412.5 08:41:52.5 2340.5 95 14 7 4 22 72 72 3413.5 08:41:53.5 2700.5 95 16 23 7 24 84.5 84.5 3416 08:41:56.0 2880.5 95 17 50 9 15 0 0 3420 08:42:00.0 3240.5 95 19 16 19 9 35.5914647586737 35.5914647586737 3447 08:42:27.0 2880.5 95 17 25 9 15 35.7159251832657 35.7159251832657 3436 08:42:16.0 2160.5 95 13 52 1 14 42.2990516638717 42.2990516638717 3472 08:42:52.0 2520.5 95 15 7 0 20 57 57 3485 08:43:05.0 3060.5 95 18 21 5 18 55.1546256119374 55.1546256119374 3505.5 08:43:25.5 3480.5 95 20 51 29 8 99.5 99.5 3525 08:43:45.0 2340.5 95 14 15 4 22 49.2666795560478 49.2666795560478 3533.5 08:43:53.5 2880.5 95 17 23 0 23 64.5 64.5 3547 08:44:07.0 3240.5 95 19 20 11 16 74.5 74.5 3581 08:44:41.0 2700.5 95 16 42 7 24 28.303029537133 28.303029537133 3589 08:44:49.0 2520.5 95 15 15 0 20 14.7797399089687 14.7797399089687 3615 08:45:15.0 3060.5 95 18 50 5 18 24.6661945810962 24.6661945810962 3643 08:45:43.0 2700.5 95 16 24 7 24 31.0556529278476 31.0556529278476 3670 08:46:10.0 3060.5 95 18 25 5 18 10.8625354505967 10.8625354505967 3676.5 08:46:16.5 3480.5 95 20 16 29 8 12.1156619534737 12.1156619534737 3687 08:46:27.0 3240.5 95 19 21 11 16 20.293115927708 20.293115927708 3701.5 08:46:41.5 2880.5 95 17 42 0 23 22.8139938790113 22.8139938790113 3679.5 08:46:19.5 2340.5 95 14 52 4 22 27.6033627495755 27.6033627495755 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 273 3713 I 08:46:53.0 2700.5 95 16 7 1 21 62 62 3719 | 08:46:59.0 2520.5 95 15 52 0 20 15.1548953894126 15.1548953894126 3746 i 08:47:26.0 3720.5 95 21 51 21 8 79.5 79.5 3772.5 i 08:47:52.5 3060.5 95 18 23 8 23 84.5 84.5 3775.5 j 08:47:55.5 3480.5 95 20 20 14 19 89.5 89.5 3758.5 | 08:47:38.5 2880.5 95 17 24 0 23 15.6802729934569 15.6802729934569 3784.5 I 08:48:04.5 3240.5 95 19 50 11 16 29.2068057900778 29.2068057900778 3822.5 I 08:48:42.5 2700.5 95 16 15 1 21 21.189326216139 21.189326216139 3845 i 08:49:05.0 3240.5 95 19 25 11 16 0 0 3834 I 08:48:54.0 2880.5 95 17 7 2 16 52 52 ! 3894.5 I 08:49:54.5 3240.5 95 19 23 3 25 77 77 3899.5 ) 08:49:59.5 3720.5 95 21 16 21 8 37.0796820626057 37.0796820626057 3926 I 08:50:26.0 3480.5 95 20 21 14 19 12.2899039012735 12.2899039012735 3939.5 i 08:50:39.5 3060.5 95 18 42 8 23 34.7271146053383 34.7271146053383 3945.5 I 08:50:45.5 2880.5 95 17 15 2 16 26.6992100535395 26.6992100535395 3959 I 08:50:59.0 2700.5 95 16 52 1 21 31.5222598313149 31.5222598313149 4010.5 ! 08:51:50.5 3480.5 95 20 50 14 19 17.5260660874827 17.5260660874827 3986 i 08:51:26.0 3960.5 95 22 51 17 8 69.5 69.5 4008 ! 08:51:48.0 3060.5 95 18 24 8 23 32.2580614143162 32.2580614143162 4026 5 08:52:06.5 3720.5 95 21 20 15 16 84.5 84.5 4057 : 08:52:37.0 3480.5 95 20 25 14 19 20.1878627712418 20.1878627712418 4079 08:52:59.0 2880.5 95 17 52 2 16 13.5816157059634 13.5816157059634 4062 i 08:52:42.0 3240.5 95 19 42 3 25 26.625886904377 26.625886904377 4072 ! 08:52:52.0 3060.5 95 18 7 1 22 64.5 64.5 4131 I 08:53:51.0 3480.5 95 20 23 5 29 92 92 4123.5 I 08:53:43.5 3240.5 95 19 24 3 25 25.9107241235012 25.9107241235012 4128 i 08:53:48.0 3960.5 95 22 16 17 8 23.8041358846617 23.8041358846617 4175 I 08:54:35.0 3720.5 95 21 21 15 16 21.6590086922245 21.6590086922245 i 4185.5 i 08:54:45.5 3060.5 95 18 15 1 22 53.3741020803076 53.3741020803076 4194.5 I 08:54:54.5 3240.5 95 19 7 1 19 57 57 4226.5 ! 08:55:26.5 4200.5 95 23 51 12 8 57 57 4272 i 08:56:12.0 3720.5 95 21 50 15 16 9.47203578202862 9.47203578202862 4241 ! 08:55:41.0 3960.5 95 22 20 16 15 84.5 84.5 4298.5 i 08:56:38.5 3480.5 95 20 42 5 29 16.5083276733241 16.5083276733241 4306 i 08:56:46.0 3240.5 95 19 15 1 19 19.2484204738052 19.2484204738052 4311.5 i 08:56:51.5 3720.5 95 21 25 15 16 25.0472432560491 25.0472432560491 4340 I 08:57:20.0 3060.5 95 18 52 1 22 36.2744676634914 36.2744676634914 4352 I 08:57:32.0 3480.5 95 20 24 5 29 3.74275448956048 3.74275448956048 4361 | 08:57:41.0 4200.5 95 23 16 12 8 23.6964912551724 23.6964912551724 4375.5 08:57:55.5 3720.5 95 21 23 11 27 102 102 4389.5 : 08:58:09.5 3960.5 95 22 21 16 15 12.9703745153732 12.9703745153732 4433.5 I 08:58:53.5 3480.5 95 20 7 4 24 77 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 274 4439.5 08:58:59.5 3240.5 95 19 52 1 19 26.1875991525283 26.1875991525283 4477 08:59:37.0 3960.5 95 22 50 16 15 36.3476847942022 36.3476847942022 4465 08:59:25.0 4440.5 95 24 51 21 8 79.5 79.5 4479 08:59:39.0 4200.5 95 23 20 13 14 74.5 74.5 4539 09:00:39.0 3720.5 95 21 42 11 27 30.372170229074 30.372170229074 4543 09 30:43.0 3960.5 95 22 25 16 15 14.0515556003056 14.0515556003056 4544.5 09:00:44.5 3480.5 95 20 15 4 24 32.0448772867643 32.0448772867643 4603 09:01:43.0 3720.5 95 21 24 11 27 31.1350461911719 31.1350461911719 4613 09:01:53.0 3960.5 95 22 23 3 26 79.5 79.5 4620.5 09:02:00.5 4440.5 95 24 16 21 8 15.7389234860793 15.7389234860793 4621.5 09:02:01.5 4200.5 95 23 21 13 14 12.2836130248426 12.2836130248426 4673.5 09:02:53.5 3720.5 95 21 7 1 27 77 77 4677.5 09:02:57.5 3480.5 95 20 52 4 24 39.279605069313 39.279605069313 4706 09:03:26.0 4680.5 95 25 51 20 8 77 77 4712 09:03:32.0 4200.5 95 23 50 13 14 4.15574188593224 4.15574188593224 4730.5 09:03:50.5 4440.5 95 24 20 12 16 77 77 4748.5 09:04:08.5 4200.5 95 23 25 13 14 5.06061638903156 5.06061638903156 4785 09:04:45.0 3720.5 95 21 15 1 27 17.0526883141438 17.0526883141438 4781 09:04:41.0 3960.5 95 22 42 3 26 8.0156513865455 8.0156513865455 4833.5 09:05:33.5 3960.5 95 22 24 3 26 17.1927505673281 17.1927505673281 4855.5 09:05:55.5 4200.5 95 23 23 7 23 82 82 4856.5 09:05:56.5 4680.5 95 25 16 20 8 5.29336716021017 5.29336716021017 4876 09:06:16.0 4440.5 95 24 21 12 16 16.2282314832981 16.2282314832981 4919 09:06:59.0 3720.5 95 21 52 1 27 22.3055422753444 22.3055422753444 4914 09:06:54.0 3960.5 95 22 7 2 21 64.5 64.5 4951 09:07:31.0 4680.5 95 25 20 11 16 74.5 74.5 4968.5 09:07:48.5 4440.5 95 24 50 12 16 0 0 4947.5 09:07:27.5 4920.5 95 26 51 16 8 67 67 5001.5 09:08:21.5 4440.5 95 24 25 12 16 20.9663978509401 20.9663978509401 5021.5 09:08:41.5 4200.5 95 23 42 7 23 0 0 5030 09:08:50.0 3960.5 95 22 15 2 21 12.3463063881264 12.3463063881264 5076 09:09:36.0 4200.5 95 23 24 7 23 12.6588642997845 12.6588642997845 5089 09:09:49.0 4680.5 95 25 21 11 16 29.6445976483914 29.6445976483914 5090.5 09:09:50.5 4920.5 95 26 16 16 8 20.4897853294716 20.4897853294716 5094.5 09:09:54.5 4440.5 95 24 23 5 26 84.5 84.5 5158.5 09:10:58.5 3960.5 95 22 52 2 21 3.77946960685088 3.77946960685088 5155 09:10:55.0 4200.5 95 23 7 1 21 62 62 5185 09:11:25.0 5160.5 95 27 51 20 8 77 77 5193.5 09 11:33.5 4680.5 95 25 50 11 16 19.6499393782459 19.6499393782459 5202.5 09 11:42.5 4920.5 95 26 20 15 15 82 82 5244 09:12:24.0 4680.5 95 25 25 11 16 16.5429576075664 16.5429576075664 5264.5 09:12:44.5 4440.5 95 24 42 5 26 20.9501583037169 20.9501583037169 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 275 5270 09:12:50.0 4200.5 95 23 15 1 ! 21 25.3010242162322 25.3010242162322 5320.5 09:13:40.5 4440.5 95 24 24 5 I 26 24.0670833095815 24.0670833095815 5333 09:13:53.0 4680.5 95 25 23 5 ; 25 82 82 5335 09:13:55.0 5160.5 95 27 16 20 i 8 8.99609205785535 8.99609205785535 5349.5 09:14:09.5 4920.5 95 26 21 15 I 15 32.580970474208 32.580970474208 5395.5 09:14:55.5 4440.5 95 24 7 1 ! 21 62 62 5400.5 09:15:00.5 4200.5 95 23 52 1 i 21 13.61664373854 13.61664373854 5433 09:15:33.0 5160.5 95 27 20 14 I 16 82 82 5426.5 09:15:26.5 5400.5 95 28 51 23 I 8 84.5 84.5 5460.5 09:16:00.5 4920.5 95 26 50 15 j 15 8.74029392445983 8.74029392445983 5500.5 09:16:40.5 4680.5 95 25 42 5 ! 25 9.18784937144202 9.18784937144202 5500.5 09:16:40.5 4920.5 95 26 25 15 i 15 25.3067173453278 25.3067173453278 5510 09:16:50.0 4440.5 95 24 15 1 I 21 19.488750675755 19.488750675755 5588 09:18:08.0 5400.5 95 28 16 23 ; 8 0.10553875567286 0.10553875567286 5554 09:17:34.0 4680.5 95 25 24 5 I 25 11.3271640958308 11.3271640958308 5574.5 09:17:54.5 4920.5 95 26 23 5 ; 25 82 82 5578.5 09:17:58.5 5160.5 95 27 21 14 i 16 14.5694630291844 14.5694630291844 5634 09:18:54.0 4680.5 95 25 7 1 i 21 62 62 5641 09:19:01.0 4440.5 95 24 52 1 I 21 28.9065577999884 28.9065577999884 5666 09:19:26.0 5640.5 95 29 51 19 I 8 74.5 74.5 5666.5 09:19:26.5 5160.5 95 27 50 14 I 16 0 0 5682.5 09:19:42.5 5400.5 95 28 20 17 i 17 92 92 5741 09:20:41.0 4920.5 95 26 42 5 i 25 12.8237554008622 12.8237554008622 5697 09:19:57.0 5160.5 95 27 25 14 I 16 6.39417429101949 6.39417429101949 i 5745.5 09:20:45.5 4680.5 95 25 15 1 i 21 14.9165469343161 14.9165469343161 5795.5 09:21:35.5 4920.5 95 26 24 5 ! 25 22.9048427862369 22.9048427862369 5804.5 09:21:44.5 5640.5 95 29 16 0 ! 0 0 0 5812 09:21:52.0 5160.5 95 27 23 8 26 92 92 5843 09:22:23.0 5400.5 95 28 21 17 17 3.2860347695154 3.2860347695154 5879 09:22:59.0 4680.5 95 25 52 1 21 37.8215949947311 37.8215949947311 5874 09:22:54.0 4920.5 95 26 7 0 ! 21 59.5 59.5 5891.5 09:23:11.5 5640.5 95 29 20 8 16 67 67 5906 09:23:26.0 5880.5 95 30 51 12 I 8 57 57 5924.5 09:23:44.5 5400.5 95 28 50 17 : 17 18.4728386632098 18.4728386632098 5985 09:24:45.0 4920.5 95 26 15 0 21 23.7798581361778 23.7798581361778 5975 09:24:35.0 5400.5 95 28 25 17 i 17 26.1032905974223 26.1032905974223 5980.5 09:24:40.5 5160.5 95 27 42 8 : 26 3.44662979712462 3.44662979712462 6022.5 09:25:22.5 5640.5 95 29 21 8 16 4.42756581463529 4.42756581463529 6034 ^ 09:25:34.0 5160.5 95 27 24 8 26 27.1119473786949 27.1119473786949 6035.5 09:25:35.5 5880.5 95 30 16 12 8 13.9237116884215 13.9237116884215 6055.5 09:25:55.5 5400.5 95 28 23 7 29 97 97 6102 09:26:42.0 5640.5 95 29 50 8 I 16 23.9712883692326 23.9712883692326 6114 09:26:54.0 5160.5 95 27 7 3 I 24 74.5 74.5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 276 6120 09:27:00.0 4920.5 95 26 52 0 21 19.6718139861789 19.6718139861789 6138 09:27:18.0 5880.5 95 30 20 7 14 59.5 59.5 6156 09:27:36.0 5640.5 95 29 25 8 16 38.0878146071646 38.0878146071646 6146.5 09:27:26.5 6120.5 95 31 51 5 8 39.5 39.5 6219.5 09:28:39.5 5160.5 95 27 15 3 24 6.66599889247703 6.66599889247703 6222 09:28:42.0 5400.5 95 28 42 7 29 10.7026956547843 10.7026956547843 6250 09:29:10.0 5880.5 95 30 21 0 0 0 0 6276 09:29:36.0 5400.5 95 28 24 7 29 13.4475356103968 13.4475356103968 6264 09:29:24.0 6120.5 95 31 16 5 8 18.2622089357901 18.2622089357901 6293.5 09:29:53.5 5640.5 95 29 23 8 23 84.5 84,5 6326 09:30:26.0 5160.5 95 27 52 3 24 31.940801226161 31.940801226161 6323.5 09:30:23.5 5880.5 95 30 50 0 0 13.2403125421837 13.2403125421837 6355 09:30:55.0 5400.5 95 28 7 3 25 77 77 6366.5 09:31:06.5 5880.5 95 30 25 0 0 4.59490935936899 4.59490935936899 6378.5 09:31:18.5 6120.5 95 31 20 10 12 62 62 6387.5 09:31:27.5 6360.5 95 32 51 8 8 47 47 6412.5 09:31:52.5 5880.5 95 30 23 3 20 64.5 64.5 6465 09:32:45.0 5400.5 95 28 15 3 25 31.1792059609331 31.1792059609331 6459.5 09:32:39.5 5640.5 95 29 42 8 23 10.5236023096192 10.5236023096192 6510.5 09:33:30.5 6120.5 95 31 21 10 12 17.7938943048758 17.7938943048758 6512.5 09:33:32.5 6360.5 95 32 16 8 8 16.9861651926325 16.9861651926325 6513.5 09:33:33.5 5640.5 95 29 24 8 23 3.92747389807214 3.92747389807214 6579 09:34:39.0 5880.5 95 30 42 3 20 15.4358496228558 15.4358496228558 6601 09:35:01.0 5400.5 95 28 52 3 25 21.8043619229966 21.8043619229966 6607.5 09:35:07.5 6120.5 95 31 50 10 12 33.125921396223 33.125921396223 6621.5 09:35:21.5 6360.5 95 32 20 9 12 59.5 59.5 6593.5 09:34:53.5 5640.5 95 29 7 4 21 69.5 69.5 6626 09:35:26.0 6600.5 95 33 51 9 8 49.5 49.5 6633 09:35:33.0 5880.5 95 30 24 3 20 23.2027227829131 23.2027227829131 6673.5 09:36:13.5 6120.5 95 31 25 10 12 0.142594699579 0.142594699579 6705 09:36:45.0 5640.5 95 29 15 4 21 23.5039948765551 23.5039948765551 6713.5 09:36 53 5 5B80.5 95 30 7 0 16 47 47 6750.5 09:37:30.5 6600.5 95 33 16 9 8 9.2634927830178 9.2634927830178 6776.5 09:37:56.5 6120.5 95 31 23 16 19 94.5 94.5 6747.5 09:37:27.5 6360.5 95 32 21 9 12 18.2914109014894 18.2914109014894 6822 09:38:42.0 5880.5 95 30 15 0 16 28.8570505891686 28.8570505891686 6839 09:38:59.0 5640.5 95 29 52 4 21 32.1118785517802 32.1118785517802 6844.5 09:39:04.5 6360.5 95 32 50 9 12 40.5978623003054 40.5978623003054 6865.5 09:39:25.5 6840.5 95 34 51 5 8 39.5 39.5 6851 09:39:11.0 6600.5 95 33 20 9 13 62 62 6917.5 09:40:17.5 6360.5 95 32 25 9 12 25.1264076828181 25.1264076828181 6958.5 09:40:58.5 5880.5 95 30 52 0 16 40.5808391462845 40.5808391462845 6947 09:40:47.0 6120.5 95 31 42 16 19 9.528538795536 9.528538795536 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 277 6976.5 09:41:16.5 6840.5 95 3.' 16 5 8 36.5970839685049 36.5970839685049 6996 09:41:36.0 6120.5 95 31 24 16 19 0 0 7015.5 09:41:55.5 6360.5 95 32 23 4 20 67 67 6978.5 09:41:18.5 6600.5 95 33 21 9 13 33.2707220892744 33.2707220892744 7075.5 09:42:55.5 6120.5 95 31 7 5 24 79.5 79.5 7087 09:43:07.0 6600.5 95 33 50 9 13 15.3258228142802 15.3258228142802 7100 09:43:20.0 6840.5 95 34 20 11 12 64.5 64.5 7106 09:43:26.0 7080.5 95 35 51 7 8 44.5 44.5 7133 09:43:53.0 6600.5 95 33 25 9 13 35.6970750506626 35.6970750506626 7182 09:44:42.0 6360.5 95 32 42 4 20 28.2816946303335 28.2816946303335 7195 09:44:55.0 6120.5 95 31 15 5 24 4.44997661544161 4.44997661544161 7213 09:45:13.0 7080.5 95 35 16 0 0 0 0 7226 09:45:26.0 6840.5 95 34 21 11 12 20.0434564677048 20.0434564677048 7246 09:45:46.0 6360.5 95 32 24 4 20 19.5910203146401 19.5910203146401 7253.5 09:45:53.5 6600.5 95 33 23 9 20 79.5 79.5 7300 09:46:40.0 7080.5 95 35 20 10 12 62 62 7314 09:46:54.0 6360.5 95 32 7 2 16 52 52 7321.5 09:47:01.5 6120.5 95 31 52 5 24 19.0624231850498 19.0624231850498 7351 09:47:31.0 6840.5 95 34 25 11 12 17.5248523435433 17.5248523435433 7318 09:46:58.0 6840.5 95 34 50 11 12 3.58440917993988 3.58440917993988 7421.5 09:48:41.5 6600.5 95 33 42 9 20 2.43932507561824 2.43932507561824 7425 09:48:45.0 7080.5 95 35 21 10 12 3.00675717633991 3.00675717633991 7425.5 09:48:45.5 6360.5 95 32 15 2 16 35.1387273006741 35.1387273006741 7474.5 09:49:34.5 6600.5 95 33 24 9 20 32.9090432907553 32.9090432907553 7492 09:49:52.0 6840.5 95 34 23 6 19 69.5 69.5 7502.5 09:50:02.5 7080.5 95 35 50 10 12 19.7813045612462 19.7813045612462 7552 09:50:52.0 7080.5 95 35 25 10 12 8.84483974387476 8.84483974387476 7554.5 09:50:54.5 6600.5 95 33 7 1 20 59.5 59.5 7612.5 09:51:52.5 7080.5 95 35 23 6 20 72 72 7620.5 09:52:00.5 6360.5 95 32 52 2 16 13.1094724464612 13.1094724464612 7660.5 09:52:40.5 6840.5 95 34 42 6 19 16.0723240200001 16.0723240200001 7663.5 09:52:43.5 6600.5 95 33 15 1 20 12.1277774792325 12.1277774792325 7712.5 09:53:32.5 6840.5 95 34 24 6 19 34.7761975694417 34.7761975694417 7793.5 09:54:53.5 6840.5 95 34 7 5 18 64.5 64.5 7779 09:54:39.0 7080.5 95 35 42 6 20 36.944007232616 36.944007232616 7799.5 09:54:59.5 6600.5 95 33 52 1 20 7.50651086312656 7.50651086312656 7848.5 09:55:48.5 7080.5 95 35 24 6 20 43.3738742069109 43.3738742069109 7973.5 09:57:53.5 7080.5 95 I 35 7 0 18 52 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 278 Table D2.2: Routel -Situation 1200 1335 08:07:15.0 1320.5 1 1 1 32 1 87 P ^ 8 7 32 1466 08:09:26.0 1320.5 1 1 39 32 1 5.4576016 5.4576016 32 1512.5 08:10:12.5 1320.5 1 1 38 32 1 29.279263 29.279263 32 1583 08:11:23.0 1320.5 1 1 37 32 1 30.155894 30.155894 32 1641.5 08:12:21.5 1320.5 1 1 36 32 1 27.893874 27.893874 32 1706.5 08:13:26.5 1320.5 1 1 35 32 1 22.669224 22.669224 32 1777.5 08:14:37.5 1320.5 1 1 34 32 1 41.966357 41.966357 32 1837 08:15:37.0 1320.5 1 1 33 32 1 26.49125 26.49125 32 1935.5 08:17:15.5 1920.5 1 2 1 17 5 49.5 49.5 17 1912 08:16:52.0 1320.5 1 1 32 32 1 21.308117 21.308117 32 1959.5 08:17:39.5 1320.5 1 1 54 3 1 14.5 14.5 12 2027 08:18:47.0 1320.5 1 1 43 3 1 8.0508578 8.0508578 12 2054.5 08:19:14.5 1920.5 1 2 39 17 5 25.892111 25.892111 17 2086 08:19:46.0 1320.5 1 1 44 3 1 2.1300374 2.1300374 12 2140 08:20:40.0 1320.5 1 1 45 3 1 19.025988 19.025988 12 2189.5 08:21:29.5 1320.5 1 1 27 3 1 14.825459 14.825459 12 2166.5 08:21:06.5 1920.5 1 2 38 17 5 17.856813 17.856813 17 2257.5 08:22:37.5 1320.5 1 1 26 3 1 32.318241 32.318241 12 2244.5 08:22:24.5 1920.5 1 2 37 17 5 24.447085 24.447085 17 2296 08:23:16.0 1920.5 1 2 36 17 5 1.262845 1.262845 17 2321.5 08:23:41.5 1920.5 1 2 35 17 5 25.467909 25.467909 17 2342 08:24:02.0 1320.5 1 1 28 3 1 0.238119 0.238119 12 2363 08:24:23.0 1320.5 1 1 3 0 0 7 7 12 2390.5 08:24:50.5 1920.5 1 2 34 17 5 54.593859 54.593859 17 2423 08:25:23.0 1320.5 1 1 40 0 0 8.3886843 8.3886843 12 2463 08:26:03.0 1920.5 1 2 33 17 5 9.4538681 9.4538681 17 2502 08:26:42.0 1320.5 1 1 29 0 0 15.033406 15.033406 12 2513.5 08:26:53.5 1920.5 1 2 32 17 5 7.141502 7.141502 17 2535 08:27:15.0 2520.5 1 3 1 9 3 29.5 29.5 9 2547 08:27:27.0 1920.5 1 2 54 2 1 12 12 9 2568 08:27:48.0 1320.5 1 1 46 0 0 18.982776 18.982776 12 2599 08:28:19.0 1920.5 1 2 43 2 1 3.2858575 3.2858575 9 2608.5 08:28:28.5 2520.5 1 3 39 9 3 7.4919369 7.4919369 9 2650.5 08:29:10.5 1320.5 1 1 30 0 0 14.60863 14.60863 12 2654 08:29:14.0 2520.5 1 3 38 9 3 30.689474 30.689474 9 2653 08:29:13.0 1920.5 1 2 44 2 1 40.306499 40.306499 9 2724 08:30:24.0 2520.5 1 3 37 9 3 4.1422931 4.1422931 9 2742 08:30:42.0 1920.5 1 2 45 2 1 33.266526 33.266526 9 2743 08:30:43.0 1320.5 1 1 41 0 0 13.267352 13.267352 12 2775.5 08:31:15.5 2520.5 1 3 36 9 3 46.905667 46.905667 9 2806 08:31:46.0 1920.5 1 2 27 2 1 17.962784 17.962784 9 2827 08:32:07.0 1320.5 1 1 31 0 0 0 0 12 2842 08:32:22.0 2520.5 1 3 35 9 3 45.471238 45.471238 9 2864.5 08:32:44.5 1320.5 1 1 47 0 0 30.329177 30.329177 12 2901 08:33:21.0 1920.5 1 2 26 2 1 38.642742 38.642742 9 2913.5 08:33:33.5 2520.5 1 3 34 9 3 20.252189 20.252189 9 2951 08:34:11.0 2520.5 1 3 33 9 3 0.6925741 0.6925741 9 2956.5 08:34:16.5 1320.5 1 1 53 0 0 23.75157 23.75157 12 2991 08:34:51.0 2520.5 1 3 32 9 3 35.313859 35.313859 9 3001.5 08:35:01.5 1920.5 1 2 28 2 1 20.597245 20.597245 9 3042.5 08:35:42.5 1920.5 1 2 3 0 0 7 7 17 3085.5 08:36:25.5 1920.5 1 2 40 0 0 22.666966 22.666966 17 3051.5 08:35:51.5 2520.5 1 3 54 10 0 32 32 10 3112.5 08:36:52.5 2520.5 1 3 43 10 0 18.541121 18.541121 10 3135 08:37:15.0 3120.5 1 4 1 12 1 37 37 12 3139.5 08:37:19.5 1920.5 1 2 29 0 0 30.632116 30.632116 17 3192 08:38:12.0 2520.5 1 3 44 10 0 31.79721 31.79721 10 3226.5 08:38:46.5 1920.5 1 2 46 0 0 17.940633 17.940633 17 3254.5 08:39:14.5 3120.5 1 4 39 12 1 46.623965 46.623965 12 3284.5 08:39:44.5 2520.5 1 3 45 10 0 30.344633 30.344633 10 3315 08:40:15.0 1920.5 2 30 0 0 3.7213291 3.7213291 17 3345 08:40:45.0 2520.5 1 3 27 10 0 29.535785 29.535785 10 3374.5 08:41:14.5 3120.5 1 4 38 12 1 47.244925 47.244925 12 3406.5 08:41:46.5 1920.5 1 2 41 0 0 33.994919 33.994919 17 3428.5 08:42:08.5 2520.5 1 3 26 10 0 0 0 10 3445.5 08:42:25.5 3120.5 1 4 37 12 1 35.577628 35.577628 12 3450.5 08:42:30.5 2520.5 1 3 28 10 0 26.298122 26.298122 10 3470.5 08:42:50.5 1920.5 1 2 31 0 0 22.255488 22.255488 17 3500 08:43:20.0 2520.5 1 3 3 7 0 24.5 24.5 21 3529.5 08:43:49.5 1920.5 1 2 47 0 0 31.603019 31.603019 17 3545.5 08:44:05.5 3120.5 1 4 36 12 1 21.006394 21.006394 12 3587 08:44:47.0 3120.5 1 4 35 12 1 27.860643 27.860643 12 3562.5 08:44:22.5 2520.5 1 3 40 7 0 31.212293 31.212293 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 279 3625.5 08:45:25.5 2520.5 1 3 29 7 0 10.208356 10.208356 21 3645.5 08:45:45.5 3120.5 1 4 34 12 1 39.903886 39.903886 12 3658.5 08:45:58.5 1920.5 1 2 53 0 0 60.452303 60.452303 17 3703 08:46:43.0 3120.5 1 4 33 12 1 0 0 12 3706 08:46:46.0 2520.5 1 3 46 7 0 25.125668 25.125668 21 3729.5 08:47:09.5 3120.5 1 4 32 12 1 42.42492 42.42492 12 3735 08:47:15.0 3720.5 1 5 1 14 3 42 42 14 3790.5 08:48:10.5 2520.5 1 3 30 7 0 46.219653 46.219653 21 3797 08:48:17.0 3120.5 1 454 4 0 17 17 4 3843 08:49:03.0 3120.5 1 4 43 4 0 17.47007 17.47007 4 3854.5 08:49:14.5 3720.5 1 5 39 14 3 21.080692 21.080692 14 3882.5 08:49:42.5 2520.5 1 3 41 7 0 44.69781 44.69781 21 3885.5 08:49:45.5 3120.5 1 4 44 4 0 7.3980621 7.3980621 4 3928 08:50:28.0 3720.5 1 5 38 14 3 20.545554 20.545554 14 3941 08:50:41.0 3120.5 1 4 45 4 0 41.351083 41.351083 4 3949 08:50:49.0 2520.5 1 3 31 7 0 24.846459 24.846459 21 3983 08:51:23.0 3720.5 1 5 37 14 3 47.41648 47.41648 14 4012 08:51:52.0 3120.5 1 4 27 4 0 44.485596 44.485596 4 4065 08:52:45.0 2520.5 1 3 47 7 0 12.064501 12.064501 21 4074 08:52:54.0 3720.5 1 5 36 14 3 10.094787 10.094787 14 4117.5 08:53:37.5 3120.5 1 4 26 4 0 16.390518 16.390518 4 4104 08:53:24.0 3720.5 1 5 35 14 3 34.430367 34.430367 14 4157.5 08:54:17.5 2520.5 1 3 53 7 0 19.801833 19.801833 21 4212 08:55:12.0 3120.5 1 4 28 4 0 20.504793 20.504793 4 4176 08:54:36.0 3720.5 1 5 34 14 3 42.966766 42.966766 14 4236.5 08:55:36.5 3720.5 1 5 33 14 3 39.467628 39.467628 14 4269.5 08:56:09.5 3120.5 1 4 3 20 0 57 57 20 4311 08:56:51.0 3720.5 1 5 32 14 3 16.461351 16.461351 14 4348.5 08:57:28.5 3120.5 1 4 40 20 0 13.757 13.757 20 4352.5 08:57:32.5 3720.5 1 5 54 4 1 17 17 4 4397 08:58:17.0 3120.5 1 4 29 20 0 0 0 20 4424 08:58:44.0 3720.5 1 5 43 4 1 43.594182 43.594182 4 4437 08:58:57.0 3120.5 1 4 46 20 0 40.25191 40.25191 20 4455.5 08:59:15.5 4440.5 1 6 1 9 4 29.5 29.5 9 4489.5 08:59:49.5 3720.5 1 5 44 4 1 42.469772 42.469772 4 4512 09:00:12.0 3120.5 1 4 30 20 0 50.600424 50.600424 20 4530 09:00:30.0 4440.5 1 6 39 9 4 28.765089 28.765089 9 4601.5 09:01:41.5 3720.5 1 5 45 4 1 8.7762561 8.7762561 4 4598 09:01:38.0 4440.5 1 6 38 9 4 9.4041155 9.4041155 9 4603 09:01:43.0 3120.5 1 4 41 20 0 19.39257 19.39257 20 4655.5 09:02:35.5 4440.5 1 6 37 9 4 58.193015 58.193015 9 4641 09:02:21.0 3720.5 1 5 27 4 1 18.189548 18.189548 4 4692.5 09:03:12.5 3720.5 1 5 26 4 1 21.816198 21.816198 4 4670 09:02:50.0 3120.5 1 4 31 20 0 34.651951 34.651951 20 4739 09:03:59.0 3120.5 1 4 47 20 0 10.047024 10.047024 20 4756.5 09:04:16.5 4440.5 1 6 36 9 4 19.992113 19.992113 9 4762.5 09:04:22.5 3720.5 1 5 28 4 1 27.582663 27.582663 4 4833 09:05:33.0 3120.5 1 4 53 20 0 38.131512 38.131512 20 4797 09:04:57.0 4440.5 1 6 35 9 4 27.719453 27.719453 9 4812.509:05:12.53720.5 1 5 3 0 0 7 7 0 4850.5 09:05:50.5 4440.5 1 6 34 9 4 25.145266 25.145266 9 4842 09:05:42.0 3720.5 1 5 40 0 0 45.733809 45.733809 0 4896 09:06:36.0 4440.5 1 6 33 9 4 23.289467 23.289467 9 4947 09:07:27.0 3720.5 1 5 29 0 0 23.412439 23.412439 0 4972 09:07:52.0 4440.5 1 6 32 9 4 30.372853 30.372853 9 5026 09:08:46.0 3720.5 1 5 46 0 0 16.464009 16.464009 0 5028.5 09:08:48.5 4440.5 1 6 54 1 0 9.5 9.5 1 5088.5 09:09:48.5 4440.5 1 6 43 1 0 5.0112064 5.0112064 1 5111 09:10:11.0 3720.5 1 5 30 0 0 26.937431 26.937431 0 5147.5 09:10:47.5 4440.5 1 6 44 1 0 19.51663 19.51663 1 5211.5 09:11:51.5 4440.5 1 6 45 1 0 16.170945 16.170945 1 5203 09:11:43.0 3720.5 1 5 41 0 0 8.7140979 8.7140979 0 5238.5 09:12:18.5 3720.5 1 5 31 0 0 30.991937 30.991937 0 5259 09:12:39.0 4440.5 1 6 27 1 0 36.61299 36.61299 1 5320 09:13:40.0 3720.5 1 5 47 0 0 39.178277 39.178277 0 5331 09:13:51.0 4440.5 1 6 26 1 0 10.976648 10.976648 1 5364.5 09:14:24.5 4440.5 1 6 28 1 0 24.464754 24.464754 1 5411.5 09:15:11.5 4440.5 1 6 3 2 0 12 12 2 5446.5 09:15:46.5 4440.5 1 6 40 2 0 19.664625 19.664625 2 5459 09:15:59.0 3720.5 1 5 53 0 0 4.7647244 4.7647244 0 5514.5 09:16:54.5 4440.5 1 6 29 2 0 25.633836 25.633836 2 5535.5 09:17:15.5 5520.5 1 7 1 15 4 44.5 44.5 15 5575.5 09:17:55.5 4440.5 1 6 46 2 0 25.972412 25.972412 2 5654 09:19:14.0 5520.5 1 7 39 15 4 12.325186 12.325186 15 5655 09:19:15.0 4440.5 1 6 30 2 0 39.763651 39.763651 2 5714.5 09:20:14.5 5520.5 1 7 38 15 4 40.036219 40.036219 15 5745 09:20:45.0 4440.5 1 6 41 2 0 41.726113 41.726113 2 5783 09:21:23.0 5520.5 1 7 37 15 4 18.74583 18.74583 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 280 5809.5 09:21:49.5 4440.5 1 6 31 2 0 20.321632 20.321632 2 5829 09:22:09.0 5520.5 1 7 36 15 4 30.83984 30.83984 15 5866 09:22:46.0 4440.5 1 6 47 2 0 15.730392 15.730392 2 5879.5 09:22:59.5 5520.5 1 7 35 15 4 22.113042 22.113042 15 5927.5 09:23:47.5 5520.5 1 7 34 15 4 14.155327 14.155327 15 5957.5 09:24:17.5 4440.5 1 53 2 0 37.11808 37.11808 2 5974 09:24:34.0 5520.5 1 7 33 15 4 25.927102 25.927102 15 6051 09:25:51.0 5520.5 1 7 32 15 4 19.132478 19.132478 15 6095.5 09:26:35.5 5520.5 1 7 54 3 1 14.5 14.5 3 6164.5 09:27:44.5 5520.5 1 7 43 3 1 52.390536 52.390536 3 6238.5 09:28:58.5 5520.5 1 7 44 3 1 0 0 3 6281.5 09:29:41.5 5520.5 1 7 45 3 1 6.4722438 6.4722438 3 6318 09:30:18.0 5520.5 1 7 27 3 1 18.891125 18.891125 3 6385 09:31:25.0 5520.5 1 7 26 3 1 8.2249188 8.2249188 3 6415 09:31:55.0 5520.5 1 7 28 3 1 27.883884 27.883884 3 6462.5 09:32:42.5 5520.5 1 7 3 4 0 17 17 4 6502.5 09:33:22.5 5520.5 1 7 40 4 0 26.902577 26.902577 4 6569 09:34:29.0 5520.5 1 7 29 4 0 8.6644114 8.6644114 4 6645.5 09:35:45.5 5520.5 1 7 46 4 0 15.449596 15.449596 4 6675 09:36:15.0 6660.5 1 1 16 2 47 47 16 6735.5 09:37:15.5 5520.5 1 7 30 4 0 49.522318 49.522318 4 6795 09:38:15.0 6660.5 1 39 16 2 25.28876 25.28876 16 6826.5 09:38:46.5 5520.5 1 7 41 4 0 21.891848 21.891848 4 6897 09:39:57.0 6660.5 1 38 16 2 13.235967 13.235967 16 6907 09:40:07.0 5520.5 1 7 31 4 0 17.445411 17.445411 4 6960 09:41:00.0 5520.5 1 7 47 4 0 12.745853 12.745853 4 6980.5 09:41:20.5 6660.5 1 8 36 16 2 25.938058 25.938058 16 6940.5 09:40:40.5 6660.5 1 8 37 16 2 8.755851 8.755851 16 7027 09:42:07.0 6660.5 1 8 35 16 2 30.233526 30.233526 16 7080.5 09:43:00.5 5520.5 1 7 53 4 0 37.665976 37.665976 4 7089 09:43:09.0 6660.5 1 8 34 16 2 24.391749 24.391749 16 7131.5 09:43:51.5 6660.5 1 8 33 16 2 1.7780939 1.7780939 16 7160.5 09:44:20.5 6660.5 1 8 32 16 2 29.271663 29.271663 16 7216 09:45:16.0 6660.5 1 8 54 8 0 27 27 8 7273 09:46:13.0 6660.5 1 8 43 8 0 5.1971143 5.1971143 8 7306.5 09:46:46.5 6660.5 1 8 44 8 0 25.060252 25.060252 8 7381 09:48:01.0 6660.5 1 8 45 8 0 11.438539 11.438539 8 7423 09:48:43.0 6660.5 1 8 27 8 0 11.162477 11.162477 8 7462.5 09:49:22.5 6660.5 1 8 26 8 0 25.012257 25.012257 8 7514.5 09:50:14.5 6660.5 1 8 28 8 0 11.078448 11.078448 8 7549 09:50:49.0 6660.5 1 8 3 3 0 14.5 14.5 3 7586.5 09:51:26.5 6660.5 1 8 40 3 0 34.01282 34.01282 3 7680 09:53:00.0 6660.5 1 8 29 3 0 29.961759 29.961759 3 7738 09:53:58.0 6660.5 1 8 46 3 0 18.006331 18.006331 3 7811 09:55:11.0 6660.5 1 8 30 3 0 20.333498 20.333498 3 7815 09:55:15.0 7800.5 1 9 1 11 2 34.5 34.5 11 7903 09:56:43.0 6660.5 1 8 41 3 0 31.899362 31.899362 3 7934.5 09:57:14.5 7800.5 1 9 39 11 2 13.838277 13.838277 11 7971 09:57:51.0 6660.5 1 8 31 3 0 29.420716 29.420716 3 7994 09:58:14.0 7800.5 1 9 38 11 2 26.232242 26.232242 11 8035.5 09:58:55.5 6660.5 1 8 47 3 0 35.974127 35.974127 3 8062.5 09:59:22.5 7800.5 1 9 37 11 2 40.146187 40.146187 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix D3 1. General This Appendix presents the basic concepts of the Kalman filter (KF) technique, and the applications of this technique to bus running time prediction developed by Reihoundt (1997), Farhan (2002) and Sahalaby et al. (2004). 2. Fundamental Concepts of Kalman Filter Technique To illustrate how the Kalman filter works, Maybeck (1979, pp 8-15) presented a very clear example, as follows. Suppose that a measurement of any one-dimensional location of a moving point is taken to be z/ at time tj and the standard deviation sdzl. Thus, the conditional probability of x(ti), the position at time t2 conditioned on the observed value of the measurement being z/ is normally distributed with mean zy and standard deviation of sdz2. Based on the conditional probability density, the best estimate of the point location A x(t1) = z] (D3.1) And the variance of the error in estimate is sd1x(tx) = sd1A (D3.2) Suppose that there is another more accurate measurement of the point location z2 of the true position but taken at time t2=ti with new standard variation sdz2. 281 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 282 With two measurements availabble for the same point location, the problem is how to give the best prediction x(t2), given t2=tj and already known z/ and z2. The Kalman filter assumes that the conditional probability density of the new position at time t2 is a Gaussian density with mean p and standard deviation sd where /i = [sd 2zl f(sdzX + s d z2 )]z, + [ s d 2x/(s d zX + s d z2 ) ] z 2 (D3.4) 1 /s d 2 = (1/sdI) + (1 /s d 2z2) (D3.5) A The best estimation is 5 of course, when x(t2) = fi or x(t2) = [sd ]2 /(sdzX + sd 2z2)\zx + [s d 2/(sd2x + sd]2)}z2= [sd], /(sd2zi + sd 2z2)](z2 - z,) Therefore, A A A x(t2) = x(tx) + K(t2 )[z2 - x(t,)] (D3.6) where: K is a Kalman Gain K (t2) = [sd2zl/(sd2zl+ sd 2z2)\ (D3.7) Now the new variance is updated as follows sd]{t2) = sdl(tx)- K { t 2)Sd 2x{tx) (D3.8) In conclusion, the system of equations says that the optimal estimate of new state A at time t2, x(t2) , is equal to the best prediction of its value before z2 is actually taken, A jc(/, ), plus a correction term of an optimal weighing value multiplied by the difference A between z2 and x(tx). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 283 i k Zi X Figure D3.1: Kalman Filter Technique- Basic Concepts (Source: Maybeck 1979) 3. The KF Model for Bus running time Prediction As mentioned in chapter 6, the Kalman Filter algorithm for bus link running time is the system of 4 main equations g(* + l) = g(^) + K47?[^gtoo, ------e(k) + VAR[dataoul]k+] + VAR[dataJk+] e(k +1) = VAR[datain ]k+i .g(k + 1) (D3.10) a(* + l) = l - g ( * + l) (D 3.ll) P(k +1) = a(k +1 ).art(k) + g(k + l).art, (k + 1) (D3.12) Where g: filter-gain Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 284 a: loop-gain e: filter error P: prediction art(k): actual running time of previous bus at instant(k) artj(k+l) : actual running time of the previous day at instant(k+l) VAR[datainJk+i = Variance of the last three days data at time step k+1. VARf dataouJk +1 = Variance of Prediction One difficulty we have to cope with is that the estimation of prediction variance. This has not been available both in the predicted form or in the form of actual observation. Farhan (2002) assumed as follows VAR[dataouJ= VAR[datai„] = VAR[localdata] Substitute VAR[localdatJ into Equations D3.9 and D3.ll, then we have Equations 6.1 and 6.2 in chapter 6. To calculate VAR[datainJ the following equation is applied VAR[data in]= E {[arti(k+l)-E[arti(k+l)]2) (D3.13) Where i = number of observed day at time step (k+1) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix D4 1. General This Appendix presents database for bus routes 1 and 95 as well as the predictions which are obtained from the developed model, the KF and the naive predictors. 2. Database for Routel Table D4.1: Database for Route 1, segment Billings-Gladstone DCODE R1 r R2.—r R4 71 3000 950 902.5 938 813 72 3000 747.5 833.5 959 788.5 90 1100 905 975 1104 925.5 52 2200 1403 1318 1323 1451 38 2200 1252 1407 1367 1401 23 1300 905 977.5 904.5 1038 22 1300 806.5 1056 1052 914.5 25 1300 912.5 880.5 977.5 886.5 39 2200 1273 1356 1239 1287 5 1200 912.5 1049 971 865.5 11 1200 986.5 1055 886 996.5 28 1300 1002 961.5 889 889.5 50 2200 1355 1424 1167 1385 53 2200 1229 1332 1357 1214 73 3000 1009 824.5 961.5 830.5 54 2200 1362 1421 1490 1460 74 3000 851.5 782 805 789 61 2100 921 974 1078 924 6 1200 915.5 841 865.5 917 70 2100 946.5 1036 1083 934.5 26 1300 863.5 812.5 995.5 956 68 2100 990 1047 998 832.5 21 1300 998 981.5 983 932.5 59 2100 1110 834 905 830 8 1200 1023 1041 997.5 910.5 285 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 286 92 1100 880 949 879.5 918.5 94 1100 862 941 972 914.5 57 2100 1065 851.5 999 941.5 4 1200 968.5 859.5 866 912.5 3 1200 852.5 912.5 792 833 16 1200 892.5 927 985.5 1018 75 3000 716.5 774.5 827.5 821 40 2200 1396 1219 1358 1281 55 2200 1166 1178 1471 1438 96 1100 1101 841 1025 934 76 3000 730.5 773.5 827.5 821 37 2200 1358 1176 1254 1276 10 1200 864.5 953.5 896 818.5 30 1300 975 856.5 1093 933.5 45 2200 1228 1366 1488 1421 35 1300 964 933 1099 1021 12 1200 861.5 848 1073 959.5 9 1200 967.5 860 925.5 1011 65 2100 749 953 806 936 58 2100 989 856.5 907.5 935.5 34 1300 860.5 944.5 761.5 1003 77 3000 1022 787.5 832 888.5 95 1100 1104 1018 980.5 1042 69 2100 771.5 848.5 869 1045 17 1200 1019 968 1016 953.5 97 1100 868 1039 995 1030 93 1100 843 1041 1089 1027 60 2100 880.5 956 962.5 1050 87 1100 842 918 907 933 29 1300 1095 946 875 952.5 78 3000 953 837 798.5 884 43 2200 1141 1231 1346 1193 46 2200 1394 1400 1250 1344 24 1300 1111 1044 1083 945.5 19 1200 930.5 896 1005 930 51 2200 1124 1358 1239 1307 62 2100 911.5 860.5 1094 833 44 2200 1362 1343 1473 1168 36 2200 1275 1397 1417 1411 32 1300 956 949 1005 1015 47 2200 1404 1223 1132 1269 66 2100 1041 1155 1103 927.5 56 2100 877.5 816.5 892.5 848.5 79 3000 660.5 775.5 857 874.5 27 1300 1023 1001 1106 1026 88 1100 995 851.5 913.5 912 74 3000 1040 988.5 1136 973.5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 287 1 1200 926.5 1036 1098 935.5 80 3000 962 1008 979 899 81 3000 831.5 781.5 823.5 876 89 1100 992.5 1055 1067 969 42 2200 1359 1416 1263 1318 82 3000 961.5 889 928 820.5 49 2200 1236 1338 1352 1345 2 1200 864.5 1032 1047 1053 98 1100 933.5 918.5 964.5 1055 67 2100 970.5 967 863 815.5 99 1100 962.5 956.5 860 922.5 64 2100 1104 1160 1122 931.5 63 2100 982.5 861 985.5 906.5 33 1300 988.5 1043 985 1005 18 1200 897 940 1108 949 83 3000 952 853.5 1035 819.5 48 2200 1471 1400 1270 1290 20 1200 860 845 1005 920 31 1300 926 982.5 983.5 1040 91 1100 965 942.5 860 929.5 100 1100 906.5 1029 1043 1049 84 3000 963 832 802.5 774 13 1200 798 977.5 871.5 917 14 1200 999.5 959.5 811 1056 101 1100 906.5 1029 1043 1049 15 1200 887 837.5 881.5 918 85 3000 1005 945 889 836 7 1200 916.5 883 788.5 938.5 Note: 1. Dcode= Date code; Scode= Situation code (Please refer to Table 6.5 for the codes); R1, R2, R3, and R4 = running time of trip 6, 5,4, and 3. 2. Predictions Resulted from the Proposed Model and Reference Predictors for Routel The results of the proposed model were returned from the Matlab program coded for the bus running time prediction module (Please see Appendix C). In this section, the example of Matlab outputs is partially presented for the sake of illustration. The predictions obtained from the developed model as well as those of reference predictors are tabulated in Tables D4.38 and D4.39. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 288 the Edist to the first second... neighbor 1.0e+003 * Columns 1 through 15 0.1962 0.2324 0.8580 0.9067 0.1797 0.3161 0.1960 0.7398 0.2573 0.2056 0.1366 0.7964 0.7745 0.2122 1.0262 Columns 16 through 30 0.1812 0.3033 0.0903 0.3319 0.2194 0.2864 0.2181 0.1666 0.2632 0.1142 0.1939 0.2129 0.0851 0.1096 0.2168 Columns 31 through 45 0.1646 0.7442 0.8952 0.2402 0.1653 0.6451 0.1759 0.3052 0.9766 0.3251 0.2869 0.1565 0.0722 0.1220 0.0928 Columns 46 through 60 0.1162 0.2561 0.1375 0.2429 0.2743 0.3509 0.2189 0.1237 0.1079 0.0720 0.7039 0.8024 0.3353 0.2171 0.7504 Columns 61 through 75 0.3240 0.8554 0.9388 0.2390 0.5855 0.4152 0.1528 0.1426 0.3497 0.1316 0.3648 0.3446 0.2312 0.1242 0.3281 Columns 76 through 90 0.8079 0.1828 0.8300 0.3196 0.2140 0.1665 0.1038 0.4333 0.2008 0.2617 0.3242 0.2753 0.7888 0.2206 0.2411 Columns 91 through 98 0.0935 0.3129 0.1728 0.1276 0.1420 0.3129 0.1055 0.1564 bandwidth = 325.1407 The matrix of Error is 1.0e+005 * Columns 1 through 15 0.0062 0.3230 1.0369 0.2701 0.0120 0.2047 0.0032 0.3904 0.0122 0.0205 0.0527 0.7882 0.2316 0.0734 0.7597 Columns 16 through 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 289 0.0526 0.0065 0.0008 0.0000 0.0457 0.0195 0.0302 0.3620 0.0571 0.0250 0.0641 0.1678 0.0202 0.0533 0.0253 Columns 31 through 45 0.4410 1.0650 0.0757 0.2722 0.3832 0.8301 0.0395 0.0128 0.2075 0.0032 0.0625 0.0149 0.3201 0.0430 0.0420 Columns 46 through 60 0.0986 0.2597 0.2452 0.0565 0.0618 0.1297 0.0380 0.0784 0.2796 0.0087 0.0470 1.0164 0.2781 0.0009 0.0205 Columns 61 through 75 0.0078 0.8448 0.3544 0.0016 1.1412 0.0690 0.0213 0.7150 0.0561 0.0518 0.0900 0.0043 0.0026 0.0878 0.0211 Columns 76 through 90 0.8060 0.0147 0.2323 0.0766 0.0004 0.0205 0.0126 0.2059 0.0277 0.0190 0.0239 0.0037 1.5853 0.0574 0.0034 Columns 91 through 99 0.0155 0.0190 0.0164 0.1868 0.0497 0.0190 0.0143 0.0662 0.0006 Cv = 1.8679e+004 The CROSS_VALIDATIOn factors 1.0e+004 * 1.0100 0.9455 1.0059 1.2357 1.5169 1.7389 1.8679 OpNei = 20 Beta = 736.0225 0.0223 -0.1607 0.3956 THE PREDCTED BUS RUNNING TIME IS 946.57 SECONDS.Program complete Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 290 h and CR Relationship 1.9 1.8 1.7 1.6 1.5 5 1.3 1.2 0.9 10 20 30 40 50 60 70 # of nearest nacjTbours Figure D4.1: An Example of the Matlab Outputs (Case 90-1100, R l) 3. Kalman Filter model’s prediction results for Routel Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 841 917 1 0 0 5 1 0 0 5 1167 905 865.5 915.5 1 0 0 5 1104 1423.5 1354.5 Observed Observed 1162 893.8 947.25 1013.19 Predict 896.75 925.5 Predict Observed 0 . 5 0.5 0.5 0.40068 0.37315 933.513 0.33946 0.41875 0.18374 883.203 975 a(k+1) 0 70630.1 e(k+1) a(k+1) e(k+1) a(k+1) Predict 973.737 2391.7 2240.81 1669.98 0.33944 864.728 2151.57 0.47704 856 e(k+1) 0 . 5 0.5 58424.1 0.5 964.5 0 . 5 0 0 . 5 0.5 0 . 5 0 0 . 5 9(k+1) 9(k+1) 0.680710.71237 1347.54 650.084 0.31929 0.28763 1059.66 1020.61 1384.5 0.81626 395.986 9(k+1) 103887 0.62685 65121.1 68848.4 0.66054 45477.2 4481.61 Var[data]n Var[data]n 1225 1675.25 0.58125 1 E + 0 6 1 E + 0 6 1E+06 1E+06 17822.3 116848 765.444 15334.7 4114.24 0.52296 A1 Var[data]n 72092.3 37313.4 15293.4 A2 A1 860.444 693.444 1979.61 6293.78 A2 A1 1133.44 3383.36 4783.4 0.5 100 870.25 1560.25 485.125 131044 76729 7225 1 E + 0 6 1 E + 0 6 A3 2669.44 A3 A3 A2 Table D4.5: Kalman Filter for test case 6-1200 Table D4.4: Kalman Filter for test case 50-2200 Table D4.3: Kalman Filter for test case 90-1100 1005 1E+06 1E+06 1059 837 2352.25 2704 12.25 2528.13 0.66056 873 1005 1E+06 1E+06 1E+06 1E+06 1124.17 133834 101867 2177.78 117851 0.59932 Average 915.333 3098.78 Average 846.667 8433.36 871.333 2040.03 6188.44 0 974 924 1057.5 161604 889 961.5 1021.67 747.111 1078.03 3620.03 912.569 art1 7 8 8 . 5 art1 Average 0 0 0 782 789 805 1077.5 art2 851.5 921 1044.67 100383 902.5 833.5 art2 996.5 889.5 917.167 986.5 1002 967 2970.25 380.25 1054.5 0 0 0 0 0 1 0 0 5 971 886 art3 865.5 art3 2 5 883 6 916.5 950 747.5 3 938.5 813 4 788.5 889 833.5 2 4 3 6 912.5 5 1049 2 3 1459.5 4 1490 5 1421 6 1361.5 8 : 1 7 8:37 8:47 8:59 8:27 time trip 8 : 1 7 8:37 8:27 8:59 8:47 8 : 1 7 8:37 8:47 8:27 time trip art3 art2 art1 8:59 time trip Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 0 5 983 1 0 0 5 1 0 0 5 863.5 995.5 812.5 Observed Observed Observed 943.5 1091 1023 917.5 910.5 988.2 998 969.75 956 1019.54 1022.54 918.75 932.5 906.528910.625 997.5 1040.5 1016.31 981 Predict 0.5 0 . 5 0.5 0.02202 0.49966 0.33234 0.27781 0.27301 0.423140.47946 970.284 0.20327 0 0 . 5 0 1389.2 0.5 17.0139 0.5 6312.93 e(k+1) a(k+1) Predict 1089.22 627.219 e(k+1) a(k+1) Predict 2491.47 0.46866 7371.96 e(k+1) a(k+1) 0.5 0.5 0.50034 g(k+D 0.97798 94.1168 0.72219 g(k+ i) 0.72699 0.53134 0.57686 1213.21 0.52054 g (k+ i) 1 E + 0 6 0 . 5 0 2778.4 96.2361 1528.07 0.5 764.035 14162.2 .6256.94 0.66766 4177.5 Var[data]n Var[data]n 3249 868.5 354.694 87.1111 34.0278 0.5 756.25 2103.13 A1 Var[data]n 1213.36 6696.69 A1 729 14400 4689 441 900 148.028 66.6944 A2 A1 4876.69 3211.11 2525.03 0.79673 2011.76 2336.11 5650.03 8649 1E+06 1E+06 1E+06 1E+06 1E+06 1E+06 1E+06 1E+06 0.5 1.36111 A3 A2 4738.03 20496.7 5525.44 12617.4 1045.44 4511.36 A3 3306.25 5017.36 23307.1 A3 A2 Table D4.8: Kalman Filter for test case 8-1200 Table D4.6: Kalman Filter for test case 26-1300 Table D4.7: Kalman Filter for test case 21-1300 962 1296 1032.5 1806.25 1190.25 5929 1498.25 927.667 44.4444 925.167 950.333 560.111 11953.8 7338.78 Average 907.667 720.028 965.167 Average Average 0 1 0 0 5 0 1005 1E+06 1E+06 1E+06 0 1 0 0 5 1083 1008.67 998 1025.5 990 933.333 173.361 946.5 art1 830 864.833 1047 834 954 832.5 1109.5 art1 0 0 917 934.5 art2 865.5 983 905 art2 art1 863.5 995.5 0 0 0 0 924 990 998 art3 934.5 956 1077.5 2 5 1036 812.5 4 1083 2 34 832.5 932 6 998 5 1047 981 2 3 4 trip trip art3 art2 trip art3 8 : 1 7 8:47 8:59 6 946.5 8:278:37 3 8 : 1 7 8:27 8:37 8:59 8:47 time 8 : 1 7 time 8:478:59 5 6 974 921 841 915.5 1036 8:27 8:37 time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 293 1 0 0 5 Observed Observed 1004 771.5 942.1 852.5 1006.86 869 1002.07 848.5 Predict 1023.25 1044.5 857.3 1065 958.75 833 959.75 941.5 956.767 999 Predict Observed 0 . 5 1 0 0 5 0.5 0.41277 831.638 912.5 0.49945 0.36288 962.047 851.5 0 8 2136.5 0.5 e(k+1) a(k+1) 5342.32 0.43253 2937.07 0.4119 5747.53 0.38949 104.472 0.5 1264.09 e(k+1) a(k+1) Predict 1804.85 3318.16 0.44341 e(k+1) a(k+1) 0.5 g (k+ i) 0.58723 0.52889 5450.27 0.47111 g(k+D 0.51434 909.841 0.48566 849.973 792 0.50055 g(k+D 16 0.5 4273 0.5 1 E + 0 6 0 . 5 1E+06 0.5 0 0.5 9414.28 0.56747 9414.28 0.61051 1E+06 0.5 0 0.5 1005 2152.63 208.944 5961.63 0.55659 Var[data]n Var[data]n Var[data]n 4096 1 E + 0 6 15006.3 4994.13 0.5881 600.25 106.778 11.1111 10305 6346.78 1768.94 A1 676 7921 1 E + 0 6 1 E + 0 6 1 E + 0 6 693.444 11736.1 693.444 11736.1 1 E + 0 6 1 E + 0 6 1056.25 2844.44 A2 765.444 1272.11 2388.28 0.63712 1521.61 4923.36 498.778 3605.69 16 16 0 18135.1 A3 A2 A1 10201 1722.25 3540.25 69.4444 348.444 A3 A3 A2 A1 Table D4.10: Kalman Filter for test case 3-1200 Table D4.9: Kalman Filter for test case 57-2100 Table D 4.ll: Kalman Filter for test case 69-2100 884 3249 1 0 0 5 1 E + 0 6 995.167 18135.1 995.167 Average 965.167 10643.4 9966.69 945.667 693.444 922.833 976.667 4011.11 949.667 2288.03 Average 0 941 972 1103.5 art1 art1 0 0 0 1005 1E+06 0 0 1005 1E+06 999 866 949 832 980.5 858 9312.25 851.5 859.5 941.5 912.5 888.5 1041.5 977.5 625 1021.5 0 0 941 862 1065 968.5 972 0 914.5 1040 997.5 879.5 910.5 918.5 914.5 914.5 1022.5 880 862 921.5 art3 art2 2 5 6 3 4 2 3 5 4 6 trip art3 art2 art1 Average 2 3 1002.5 4 761.5 trip art3 art2 8 : 1 7 8:37 8:47 8:59 8:27 8:37 8 : 1 7 8:27 8:47 8:59 time time 8:37 8:47 5 860.5 8 : 1 7 8:27 8:59 6 860.5 1021.5 1103.5 time trip Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 294 953 884 1 0 0 5 1005 Observed 907 1 0 0 5 1230.5 1141 1192.5 Observed 886.5 842 968.8 1027.25 933 996.807 918 978.75 Predict Observed 911.367 837 1089 Predict 944.5 0.5 a(k+1) Predict 0.2348 0.28364 877.553 798.5 0.48889 0.4549 977.73 1345.5 189.329 0.16113 e(k+1) 25.8681 0.5 e(k+1) a(k+1) 1779.95 835.802 3290.04 0.48925 283.124 0 0 . 5 e(k+1) a(k+1) 0 . 5 0 0 . 5 g(k+i) 0.67020.83887 949.342 0.3298 0.50221 1462.06 0.49779 947.815 g(k+D 0.7652 0.51075 0.51111 8192.53 g(k+1) 0.54510.60764 657.772 728.429 0.39236 1036.51 225.694 3559.9 0.5 1166.74 0.71636 6441.57 1 E + 0 6 0 . 5 1198.78 478.569 0.5 239.285 0.5 36 370 2809 2911.25 277.778 24284 1 E + 0 6 4011.11 1E+06 1E+06 1E+06 69.4444 200.6945402.25 51.7361434.028 0.5 9441.36 2055.11 667.361 1 E + 0 6 17336.1 106.778 16028.9 860.444220.028 1534.03 2085.44 3802.78 1206.69 A2 A1 Var[data]n A3 A2 A1 Var[data]n 1E+06 1E+06 1E+0617.3611 1E+06 0.5 0 0.5 420.25 2304 529 625 1416.5 3441.78 2272.11 61.3611 1586.69 312.111 96.6944 2193.36 Table D4.13: Kalman Filter for test case 78-3000 Table D4.14: Kalman Filter for test case 43-2200 Table D4.12: Kalman Filter for test case 87-1100 940 256 484 863.833 1016 1015.5 939.167 1 0 0 5 1 E + 0 6 860.167 900.333 0 1 0 0 5 1 E + 0 6 875 914.833 880.5 952.5 978.333 5064.69 1041 art1 Average A3 A2 A1 Var[data]n 0 1005 953 963.333 14721.8 aril Average A3 884 923.167 837 907 933 art2 art1 Average 843 1039 art2 0 0 0 875 798.5 0 0 art3 868 0 art3 art2 907 962.5 1049.5 art3 0 995968 1089 962.5 2 3 6 2 35 933 952.5 918 946 trip 4 trip 2 3 1029.5 1027 1049.5 1035.33 34.0278 8 : 1 7 8:59 6 842 1095 8:27 8:37 8:47 time 8:59 6 880.5 842 1095 8:37 4 8 : 1 7 8:27 8:47 5 956 918 946 time 8:59 8:17 8:27 8:378:47 4 5 time trip Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 0 5 1124 877.5 892.5 816.5 1306.5 1238.5 1357.5 Observed 1 0 0 5 Observed 1111 Observed 1082 1295 967.5 938.9 1286.4 1044 966.25 848.5 Predict 1174.25 945.5 1046.68 1112.71 1082.5 0.5 0.5 0 . 5 1 0 0 5 0.45984 1053.5 0.21977 a(k+1) Predict a(k+1) Predict 0.28159 0 0 e(k+1) a(k+1) 22356.6 10610.3 0.5 e(k+1) e(k+1) 0.5 0 . 5 0.5 17073.1 0.5 0.765120.54016 7577.1 23434.8 0.23488 1075.82 fl(k+1) 0.64524 18467.4 0.35476 0.54517 28525.1 0.45483 0.53881 10393.8 0.46119 1033.67 9(k+1) 9(k+1) 0.64322 34459.8 0.35678 0.71841 15959.2 1E+06 0.5 0 0.5 28621 1E+06 43385.4 34146.1 4160.69 0.78023 3246.28 21220.6 0.5 53574.2 22214.8 Var[data]n Var[data]n 50475.1 A1 Var[data]n 8587.11 52322.8 513.778 41412.3 53515.1 59454.7 A1 1 E + 0 6 1 E + 0 6 4784.03 47161.4 A2 A1 45867.4 14081.8 78322.7 0.54914 43010.1 0.45086 469.444 18906.3 900 11556.3 9903.13 73170.3 16256.3 20449 44713.3 0.5 5600.03 2721.36 3065.47 39375.8 20468.1 31565.4 73080.1 25546.7 13034 2085.44 19290.4 A3 A3 A2 43960.1 Table D4.15: Kalman Filter for test case 24-1300 Table D4.17: Kalman Filter for test case 56-2100 Table D4.16: Kalman Filter for test case 51-2200 1005 1E+06 1E+06 1E+06 1073 1005 1E+06 1E+06 1E+06 1E+06 1113.17 81986.8 1155.17 200.694 57041.4 1133.67 1108.83 1070.57 Average A3 A2 1131.33 110778 Average Average 896 0 930.5 1041 1250 1394 1162.67 1154.5 art1 0 0 1044 1394 945.5 930 1132 1102.5 1079.83 1404 1223 1269 927.5 art2 art1 1192.5 1343.5 1140 65536 2756.25 0 0 0 1005 1E+06 0 956 949 0 0 art3 art2 art1 884 1399.5 837 1230.5 1399.5 1155.67 101548 5600.03 953 1141 art3 art3 art2 798.5 1345.5 2 6 5 4 1005 3 1015.2 2 34 1343.5 1250 1082.5 1005 1112.5 5 6 1141 2 3 5 6 4 trip trip 8 : 1 7 8:27 8:59 8:37 8:47 8:59 8:37 8 : 1 7 8:27 8:47 time 8:27 8:37 8 : 1 7 8:47 8:59 time trip time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 0 5 1 0 0 5 1040 1001 1136 973.5 988.5 Observed Observed 998.7 988.5 958.5 992.1 770.3 1023 955.75 1005 939.75 1025.5 Predict Observed Predict 908.917 1042.5 925.506 957.868 940.985 0 . 5 0 . 5 1 0 0 5 0 . 5 a(k+1) 0.38643 0.27067 0.37388 0.45387 0.32249 a(k+1) Predict 0 0 . 5 19538.6 0.44901 8590.54 0.49951 995.241 985 6801.01 0.48549 938.805 1106 8966.99 e(k+1) a(k+1) e(k+1) e(k+1) 0.5 790.25 0.5 0.50049 g(k+D 9(k+1) 0.626120.55099 8053.6 0.67751 11246.6 g (k+ i) 16014 0.54191 8678.19 0.45809 1 E + 0 6 0 . 5 0 33461 0.54613 18273.9 17164.4 14614.4 0.61357 12862.6 5291.74 0.72933 3859.42 5860.69 0.5 2930.35 Var[data]n 81 1580.5 792100 792100 0.5 0 17248.4 1144.69 8773.44 13218.4 0.51451 A1 Var[data]n 1225 7685.44 3383.36 A2 27944.7 A2 A1 Var[data]n 1E+06 1E+06 1E+06 1E+06 0.5 792100 792100 10100.3 15625 600.25 3948.03 7773,36 641.778 23053.4 A3 Table D4.19: Kalman Filter for test case 27-1300 TableD4.20: Kalman Filter for test case 74-3000 Table D4.18: Kalman Filter for test case 33-1300 876 883.5 1936 920.167989.167 5.44444 128.444 16684 186.778 17644.7 66.9444 13.4444 0.5 33.4722 0.5 992.333 1284.03 859.667 32881.8 318.028 39667.4 16599.9 892.833 53978.8 16943.4 10438 35461.1 Average Average A3 A2 A1 Average A3 0 1 0 0 5 995 861 art1 art1 851.5 art1 0 0 0 1005 1E+06 1E+06 1E+06 0 0 8 9 0 art2 877.5 660.5 892.5816.5 857 775.5 950.667 915.5 57121 9801 19600 848.5 874.5 1001 1159.5 1025.5 912 937.333 art2 0 0 0 1041 860 1122 985.5 857 1106 913.5 958.833 10370 21658 2055.11 1154.5 956.5 922.5 931.5 906.5 962.5 1104 982.5 1016.33 2898.03 2 3 2 5 6 4 6 2 3 874.5 56 775.5 660.5 1023 45 1102.5 3 927.5 4 trip art3 art2 trip art3 8:27 8:37 8 : 1 7 8:59 8 : 1 7 8:47 8:47 8:59 8 : 1 7 8:27 8:37 8:47 8:59 8:37 8:27 time time time trip art3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 860 1 0 0 5 942.5 929.5 1 0 0 5 1 0 0 5 1450 1421 1322.5 1317.5 Observed Observed Observed 932.8 965 1097 966.1 962.5 Predict 910.25 922.5 965.25 928.639 956.5 0 . 5 0.5 1022.25 0 . 5 0.5 a(k+1) 0.34934 964.635 0.46183 925.926 0.428090.37741 1252.12 1106.15 a(k+1) Predict 0 e(k+1) 1204.97 0.31469 1599.59 1533.03 2547.87 0.35753 884.273 860 e(k+1) a(k+1) Predict e(k+1) 0 . 50.5 17350.8 0 52-2200 g(k+i) 0.65066 13089 0.53817 42614.5 0.64247 0.68531 9(k+1) 0.62259 g (k+i) 79184 1 E + 0 6 0 . 5 0 34701.7 20116.6 1758.28 1879.24 0.5 939.618 0.5 3965.74 10603.1 0.53371 5658.98 0.46629 3228.13 0.64148 2070.79 0.35852 2796.94 0.57191 Var[data]n Var[data]n 792100 792100 10506.3 5088.44 2462.36 50925.4 25493.4 99703.6 0.58804 58629.6 0.41196 26623.4 1906.78 792100 792100 792100 0.5 6480.25 25281 6321.25 0.5 3160.63 0.5 A2 A1 A2 A1 Var[data]n 105084 53284 8711.11 792100 792100 148482 A3 A2 A1 7 9 2 1 0 0 6162.25 3540.25 2916 30.25 860.444 2898.03 6916.69 3885.44 1708.44 10746.8 Table D4.21: Kalman Filter for test case 99-1100 Table D4.22: Kalman Filter for test case 91-1100 Table D4.23: Kalman Filter for test case 8 9 0 1086 33672.3 6561 8 9 0 1075.83 1085.67 1083.17 42780 Average 1000.33 922.833 3402.78 113.778 2272.11 842.333 0 926 1104 970.5 art1 Average A3 art1 Average A3 art1 0 0 0 1005 1E+06 1E+06 1E+06 860 920 1039.5 959 845 982.5 1055 815.5 974.5 1005 983.5 918.5933.5 967 972.5 788.5 925.5 747.5 905 867.5 6806.25 14400 1406.25 art2 833.5 975 903.667 1.36111 4923.36 0 0 0 0 0 813 950 1471 art3 1269.5 art3 art2 2 3 1053 56 1032 864.5 4 1047 964.5 863 958.167 7891.36 40.1111 9056.69 2 5 902.5 3 46 938 3 1290 2 6 5 1400 4 trip art3 art2 trip trip 8:59 8 : 1 7 8:27 8:37 8:47 8 : 1 7 8:27 8:37 8:59 8:47 time time 8:27 8:37 8 : 1 7 8:59 8:47 time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 0 5 1421 1362 1459.5 1 0 0 5 1 0 0 5 1356 1273 Observed 1286.5 1238.5 Observed Observed 1123 1137.2 1490 917.75 1404 1252 1108.84 Predict 1068 1344.681337.82 1367 1406.5 945.75 Predict Predict 0 . 5 0.5 1227.75 1400 0.5 a(k+1) 0.35281 0.28618 0.42954 0.41105 1104.51 15849.1 0.5 e(k+1) 10468.4 e(k+1) a(k+1) 24585.3 1805.55 0.35131 2354.03 3203.37 2356.02 0.34614 1004.42 e(k+1) a(k+1) 0 . 50.5 0 0 . 5 0.72275 9300.25 0.27725 g(k+D 0.57274 20677.4 0.42726 54-2200 39-2200 38-2200 0.58958 14138.5 0.41042 g (k+ i) g(k+D 1E+06 12867.8 18994.2 0.64719 12293 31698.3 1 E + 0 6 0 . 5 0 14665.3 0.71382 2783.4 0.64869 4708.075439.133603.24 0.5 0.58895 0.65386 Var[data]n Var[data]n 1 E + 0 6 136038 36102.6 35595.1 97552.1 A1 Var[data]n 37636 1E+06 1E+06 0.5 0 0.5 75900.3 23980.6 A1 8250.69 37960 40066.7 4489 5476 0.25 600.25 16727.1 156552 43830.3 0.5 21915.1 12844.4 147584 43097.6 0.57046 A2 1013.36 3580.03 7168.44 A2 A1 1 E + 0 6 1 E + 0 6 58402.8 4993.78 A3 A2 1E+06 1E+06 1E+06 1 E + 0 6 1 E + 0 6 70933.4 8402.78 920.111 4646.69 1431.36 38.0278 5402.25 Table D4.24: Kalman Filter for test case Table D4.26: Kalman Filter for test case Table D4.25: Kalman Filter for test case 978 1197.17 24753.8 981.778 1161.671193.33 28.4444 52976.7 19228.4 1 0 0 5 1128.5 28730.3 946.333 874.667 0 0 1 0 0 5 961.5 886.5 912.5 977.5 art1 Average A3 1322.5 1402.5 1018.33 73350.7 0 0 0 1 0 0 5 914.5 806.5 1104 art2 art1 Average art2 art1 Average A3 0 0 0 0 905 1038 art3 art2 977.5 1056 880.5 971.333 904.5 1052 art3 1167 1356.5 1384.5 1213.5 830.5 1142.83 art3 3 6 2 5 4 3 788.5 925.5 1450.5 1054.83 2 5 833.5 975 1317.5 1042 43472.3 2 trip 4 3 trip trip 8:27 8:47 8:59 8 : 1 7 8:37 time 8 : 1 7 8:37 4 959 8:27 8:47 8:59 6 747.5 905 time 8 : 1 7 8:37 8:47 5 1423.5 1332 824.5 8:59 6 1354.5 1228.5 1008.5 8:27 time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 0 5 1 0 0 5 1438 1254 1176 1358 1 0 0 5 1358 1177.5 1275.5 1470.5 Observed Observed Observed 913 1300 1166 913 1280.5 867.7 909.7 1396 1142.75 1308.09 1391.09 0 . 5 0 . 5 0 .5 0.5 0 . 5 a(k+1) Predict 0.400490.22494 1008.92 0.38484 905.751 1218.5 16942.9 0.41356 e(k+1) 26211.1 0.43696 e(k+1) a{k+1)39153.4 Predict 5861.65 e(k+1) a(k+1) Predict 4858.81 0 . 5 0 0 . 5 0 0.5 12076.6 0 .5 0.6445 13431.9 0.3555 5 5 -2 2 0 0 9(k+1) 0.56304 9(k+1) 0.58644 0.61021 42251.3 0.38979 1002.13 0.689950.69192 23791.8 13212.7 0.31005 0.30808 922.48 4 0 -2 2 0 0 3 7 -2 2 0 0 g(k+D 0.61516 1909.45 3104 1E+06 0.5 0 0.5 9777.4 0.59951 28891.3 2396.74 0.77506 1857.62 9717.63 90601 78493.4 69240.9 A1 Var[data]n 58000.7 24153.2 A1 Var[data]n 59211.1 78306.8 0.5 24701.4 34483.4 1667.36 9376.69 4830.25 A1 Var[data]n 5184 10816 13728 1 E + 0 6 1 E + 0 6 16986.8 A2 39534.7 60106.7 20840.7 52670.3 8040.11 3098.78 1024 11881 81225 155236 46553 27889 10302.3 72092.3 19095.6 139627 1 E + 0 6 2146.78 5112.25 60926.7 A3 A2 1694.69 3306.255826.78 16129 A3 A2 Table D4.2S: Kalman Filter for test case Table D4.29: Kalman Filter for test case Table D4.27: Kalman Filter for test case 999 1005 1E+06 1E+06 1E+06 1E+06 1005 1E+06 1E+06 1E+06 1E+06 1057 1 0 0 5 1001.5 820.5 1064.33 1039.67 491.361 47815.1 1107.67 131648 6833.78 930.667 Average A3 0 0 821 art1 773.5 art1 Average 841 934 1025 827.5 927 774.5 871.333 1100.5 730.5 art2 art1 Average 716.5 1395.5 art2 774.5 1218.5 973.333 892.5 716.5 985.5 827.5 868.333 1017.5 821 890.5 0 0 0 0 0 0 0 792 1438 art3 art2 1017.5 821 1280.5 art3 2 5 912.5 6 852.5 3 833 2 2 3 5 927 56 1177.5 1166 4 985.5 827.5 1358 trip 8 : 1 7 8:47 8:59 8:27 8:37 4 time 8 : 1 7 8 : 1 7 8:27 8:59 6 892.5 8:37 8:47 8:278:37 3 4 1470.5 8:47 8:59 time trip art3 time trip Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 300 1 0 0 5 1141 1 0 0 5 1250 1394 1345.5 1192.5 1 0 0 5 Observed Observed 1089 1167 1228 1213 977.73 969.25 1420.5 1208.151053.22 1487.5 1365.5 1098.751344.57 1343.5 1236.94 1399.5 Predict Observed 0.5 0.4549 0.39236 1036.51 1230.5 a(k+1) Predict 0.48889 0.4929 a(k+1) 0.33001 0.27724 a(k+1) Predict 0 0 . 5 0 0 . 5 239.285657.772 0.5 944.5 8192.53 e(k+1) e(k+1) 4765.03 0.5 208403 11834.6 0.31175 26821.8 e(k+1) 0 . 5 0 0 . 5 0.5 0 . 5 0.60764 728.429 4 5 -2 2 0 0 g(k+i) 4 6 -2 2 0 0 4 3 -2 2 0 0 0.5071 g(k+i) 0.669990.72276 10545.2 4742.76 g(k+D 0.64742 20766 0.35258 fE+06 1 E + 0 6 0 . 5 1 E + 0 6 478.569 0.5 30221.6 0.53654 16215 0.46346 9530.07 Var[data]n Var[data]n 6084 6562 33428 A1 1534.03 106.778 16028.9 0.51111 3802.78 1206.69 0.5451 A1 136.111 32075 28448.4 410972 12100 34163.4 A2 A1 Var[data]n 1024 1E+06 1E+06 1E+06 17248.4 43194.7 115034 3422.25 28056.3 51076 15739.3 3268.03 15792.1 96.69442193.36 860.444 220.028 312.111 2085.44 4011.11 1198.78 Table D4.30: Kalman Filter for test case Table D4.32: Kalman Filter for test case Table D4.31: Kalman Filter for test case 1005 1E+06 1E+06 1E+06 1005 1E+06 1E+06 1E+06 1009.67 860.167 Average A3 A2 1009.17 70933.4 36353.8 5725.44 53643.6 0.5 0 1 0 0 5 0 0 art1 Average A3 A2 art1 856.5 995.333 32640.4 1750.03 19274.7 17195.2 0.68825 art1 Average A3 0 875 798.5 1095 953 963.333 14721.8 17336.1 952.5 884 923.167 art2 896 1092.5 1080.83 29986.7 953.5 0 907 918 946 837 900.333 0 0 art3 art2 875946 798.5 1345.5 837 1006.33 1230.5 1004.5 1095 953 1141 1063 0 0 952.5 884 1192.5 1254 1176 1357.5 86.5 975 806.333 303785 518160 art3 art2 2 4 3 933 5 6 842 2 3 5 4 6 3 1275.5 818.5 933.5 2 5 6 trip 8:37 8 : 1 7 8:27 8:47 8:59 8 : 1 7 8:27 8:37 8:47 8:59 time trip art3 8:27 8 : 1 7 8:37 4 time trip 8:47 8:59 time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 0 5 1404 1 0 0 5 1124 1 0 0 5 1168 1362 1306.5 Observed Obaerved 1052 1082 1017.87 1223 Predict 1053.5 1357.5 997.2 Predict 0 . 5 0.5 1010.1 1269 0.36077 a(k+1) 0.39714 1109.85 1132 0.35476 a(k+1) 0.234880.45984 1075.82 1238.5 0.276140.46244 1114.430.19887 1143.51 1472.5 1342.5 a(k+1) Predict Observed 0 0 . 5 11523.2 10542.8 e(k+1) 18467.4 22356.6 0.5 967.5 23434.8 e(k+1) e(k+1) 22239.5 0.5 919 0.5 0 . 5 0 0 . 5 0 . 5 0.62364 12730.1 0.37636 9(k+1) 5 1 -2 2 0 0 4 7 -2 2 0 0 0.64524 4 4 -2 2 0 0 0.54016 fl(k+1) g(k+D 0.72386 9928.61 1 E + 0 6 0 . 5 0 16492.9 0.63923 23046.4 22246.3 0.60286 13411.3 43385.4 10849.2 0.80113 8691.68 61122.1 0.53756 32856.8 44478.9 0.5 Var[data]n 78680.3 20412.6 36163.4 A1 Var[data]n 14042.3 85849 5980.44 58402.8 45198.8 33379.3 A2 A1 A2 A1 Var[data]n 27005.4 A3 A3 Table D4.34: Kalman Filter for test case Table D4.33: Kalman Filter for test case Table D4.35: Kalman Filter for test case 1005 1E+06 1E+06 1E+06 1229.5 12769 28056.3 1005 1E+06 1E+06 1E+06 1E+06 1197.67 Average 1113.17 81986.8 4784.03 47161.4 Average 1023.17 8680.03 80277.8 0 896 949 1005 1298 30450.3 833 art1 1094 1112.5 11556.3 15876 342.25 13716.1 911.5 988.667 3383.36 18315.1 5954.69 art1 0 1044 945.5 930 1073 73170.3 16256.3 20449 44713.3 0.5 1124 1275 956 1397 1082.5 1005 1112.5 18906.3 900 11556.3 9903.13 0.76512 7577.1 art2 1306.5 1410.5 1015.2 1197.9 894.01 1357.5 860.5 1038 20164 102080 31506.3 1416.5 art2 0 0 0 0 0 0 1005 1E+06 1E+06 1E+06 1E+06 0 1250 1141 1394 930.5 1155.17 200.694 57041.4 50475.1 28621 art3 art2 art1 Average A3 A2 1362 1168 1343.5 1399.5 art3 2 3 930 56 896 930.5 4 1005 1238.5 2 6 3 4 5 2 6 5 1342.5 3 4 1472.5 trip 8 : 1 7 8:27 8:47 8:59 8:37 time 8:59 8 : 1 7 8:27 8:37 8:47 time trip art3 8 : 1 7 8:59 8:37 8:47 8:27 time trip Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 0 5 1471 1290 1318 1269.5 1415.5 1086 987 1108 1359 1138.24 1021.69 1400 1188.99 1263 1146.43 Predict Observed 0 . 5 0.5 912.25 0 . 5 1 0 0 5 a(k+1) Predict Observed 0.29927 0.40431 0.48705 0 0 1770.13 e(k+1) 2241.13 0.40485 4198.47 8756,12 0.4409 435.139 0.5 0.59569 2820.51 4 2 -2 2 0 0 4 8 -2 2 0 0 0.51295 g(k+1) e(k+1) a(k+1) 2102.57 0.70073 1473.33 5276.24 0.72674 3834.46 0.27326 Var[data]n g(k+1) 11025 3540.25 0.5 1E+06 1E+06 0.5 11342.3 15661.1 0.5591 2952.11 870.278 0.5 A1 Var[data]n 4074.69 4225 56.25 3765.63 0.59515 A2 A1 1495.11 A2 1E+06 1E+06 1E+06 1E+06 0.5 1820.44 2384.69 38.0278 A3 9441.36 28.4444 8433.36 4734.9 1111.11 9441.36 A3 Table D4.36: Kalman Filter for test case Table D4.37: Kalman Filter for test case 1005 1E+06 1E+06 Average 914.667 245.444 928.667 Average 0 1 0 0 5 1035 1042.5 3306.25 art1 819.5 924.5 6480.25 600.25 853.5 945.333 969 992.5 1066.5 890.5 11881 4489 30976 8185 1054.5 948 3600 27722.3 0 897 952 945.833 876 1107.5 art2 art1 0 0 0 0 899 985 781.5 823.5 988.5 2 3 2 3 1005 949 5 1042.5 940 6 trip art3 trip art3 art2 8 : 1 7 8:47 5 1008 781.5 8:37 4 8:59 6 962 831.5 8:27 time 8 : 1 7 8:47 8:27 8:37 4 8:59 time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.161 d3 0.2676 0.20344 0.16661 0.00305 0.20612 0.00416 0.06481 0.25418 0.09561 0.13135 0.07008 0.35374 0.01661 d2 0.0927 0.14035 0.06996 0.01036 0.26799 0.04442 0.19531 0.01574 0.04645 0.2279 0.10499 0.43098 0.19275 0.03737 0.05414 0.03381 0.06601 0.03337 0.07948 0.09154 0.00365812 0.038624178 0.0690909090.064891192 0.164236417 0.00373 0.109217986 0.24701 0.122969121 0.05226 0.136998951 0.151062106 0.203291815 0.0668436870.068586797 0.115530516 0.00985 0.135845749 0.045303867 0.0170542640.002304752 0.2113 0.02366 0.203914452 0.043812317 0.169473075 1484 Naive 1422.5 852.56 959.56 736.22 968.83 780.01 795.25 Model dl 1401.03 857 968 942 1028.22 KF. 998.74 1189.6 770.31 943.55 742.31 Model 946 856 874.4 874.29 938.89 822.44 950.32 911.27 945.54 886 845.5 917.61 893.84 931.29952.37 988.17 928.06 1090 1384.2 902.38 932.8 888.3 928.82 1104 920.21 943.17 1212.7 1039 Model 1009.84 Developed 905 1362 1132.69 997 1023 1124 1358.5 1082 1409.7 962.5 896 966.09 Actual Table D4.38: Prediction Error of the Predictors situation) Error the D4.38: (Overall ofPredictors Prediction Table 1300 988.5 1100 842 1200 915.5 1200 1022.5 1100 1100 965 1200 852.5 889.85 13001300 863.5 998 3000 953 2200 2100 877.5 2200 1354.5 1377.6 1068.3 2100 1065 941.96 2200 1141 1296 1088 2100 771.5 2200 Scode 8 6 3 87 33 78 51 56 50 57 90 91 44 99 1100 27 1300 69 24 1300 1111 26 21 43 Dcode Note: dl,d2, d3: The absolute relative error (%) ofthe Developed Model, the Kalman Filterand the Naive predictor, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 304 0.12805 0.09195 0.019037 0.043137 0.064884 0.121697 0.191266 0.237633 0.019059 0.2601 0.137112 0.349198 0.148307 0.203201 0.029947 0.203708 0.217825 0.108749 0.175216 0.145597 0.340285 0.346389 0.170125 0.080064 0.038895 0.119971 0.072727 0.044207 0.129269 0.337818 0.154724 0.25098 1562 0.117005 0.274439 1384.2 0.104456 1422.5 0.163501 1409.7 0.167201 920.21 1497.24 1332.27 0.019251 0.178895 857 780.01 0.087729 1068.31162.2 795.25 1483.46 0.042225 0.016478 1212.7 1108.1 1122.9 1028.22 1187 1052 1233.32 941.96 1227.24 909.69 1127.25 1331.72 1065 1394 1332 1471 1306.9 1086.1 1404 1124 1358.5 1082 1251.5 1306.05 1404.02 1272.5 2200 1357.5 1211 867.75 2200 1362 1132.69 997.21 2200 1402.5 1290.21 1097 1222.91 2200 1395.5 2200 1354.5 1377.61 Table D4.39: Prediction Error of the Predictors Predictors the Error (situation of 2200) Table D4.39: coded Prediction as 52 383950 2200 2200 37 57 2200 54 2200 1361.5 1259.5 51 2200 40 46 2200 4248 2200 2200 1359 1332 44 47 2200 Note: dl,d2, d3: The absolute relative error (%) ofthe Developed Model, the Kalman Filter andthe Naive Models, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 305 4. Database for Route 95 Table D4.40: Database for Route 95, segment Laurent-Mackenzie (70 simulation runs) Dcode ? S co d e R1 R2 R3 R4 54 2200 794.5 681.5 677 747 58 2200 681.5 798 662 741.5 33 1200 571 537.5 561.5 605.5 42 1200 867 747 746 686.5 47 1200 747 866.5 747 686.5 62 2200 699 672.5 673 642.5 48 1200 746 866.5 747 808.5 36 1200 746.5 755 748 686 16 3000 632 632.5 648.5 592 52 2200 781.5 785 686.5 740 3 3000 803.5 647.5 795 731.5 2 3000 673.5 666 687 633.5 7 3000 698.5 677 666.5 725 21 1200 563 638 548.5 526 28 1200 562 657 561.5 511.5 19 1200 553.5 547.5 546 613 4 3000 782 792 680.5 619 11 3000 685 753.5 797 641.5 45 1200 747 773 747.5 686 39 1200 746 746.5 643.5 698.5 5 3000 690 770.5 679.5 735.5 55 2200 685 795 677.5 737.5 38 1200 746.5 747 763 688 13 3000 668 792.5 677 728.5 64 2200 694 830.5 755 752 14 3000 689 674 683 627 10 3000 672.5 661.5 667 615.5 29 1200 559 548 563 614.5 35 1200 548 521.5 559.5 499.5 53 2200 820 694.5 671.5 743.5 17 1200 550 526 550.5 589.5 6 3000 720 810 750 700 12 3000 689.5 795.5 680 717 18 1200 550 641.5 571.5 525 31 1200 558 558.5 557 509 56 2200 805.5 699 672.5 728 8 3000 681 647.5 691.5 743 49 1200 748.5 747.5 746 686.5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 306 9 3000 679.5 804.5 682 745 59 2200 778.5 781 687 740 41 1200 748 754.5 638.5 809.5 46 1200 747 766.5 746.5 807 57 2200 671 691.5 683 734 51 2200 694.5 777.5 686 734.5 1 3000 702 676.5 676.5 737 22 1200 556 540.5 554 502 40 1200 747.5 746.5 746 708 63 2200 798 796.5 675 732 50 1200 820 788 777 790 61 2200 683 763.5 688 833.5 43 1200 748 746.5 644.5 811.5 24 1200 557 542 564 580 25 1200 566 535.5 554.5 612.5 37 1200 746 869 747.5 707.5 65 2200 685 772 686 712 44 1200 646 746.5 748.5 686.5 60 2200 686 678 681 722.5 20 1200 554 538.5 561.5 509 34 1200 553.5 657.5 554.5 514.5 26 1200 553 640 555.5 496.5 23 1200 569.5 548.5 567 609 30 1200 565.5 552.5 568.5 631 32 1200 572.5 569.5 675.5 513 27 1200 573.5 553.5 551.5 513 15 3000 655 673 648 632 66 1200 755.5 866.5 633.5 685.5 67 1200 747 746.5 747.5 809.5 68 1200 756.5 866.5 755.5 931.5 69 1200 747 746 747.5 806.5 70 1200 746.5 748.5 748 696 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 307 5. Route 95- Running time Predictions of Developed Model and the Reference Predictors The results of the developed model were returned from the Matlab program (Please see Appendix C). The predictions of the developed model as well as that of reference predictors are presented in the Table D4.62. the Edist to the first second... neighbor Columns 1 through 12 109.1627 295.7930 9.8234 118.3860 119.4288 163.0376 11.9269 184.8574 83.9673 116.9198 120.1395 112.2297 Columns 13 through 24 284.4477 277.8394 296.8063 111.2542 73.4592 26.4670 104.5490 82.0762 94.0997 17.0660 89.6284 99.5440 Columns 25 through 36 120.5664 143.5627 284.7236 354.5046 104.9976 315.9980 61.6624 85.2877 268.0340 327.9482 95.7836 124.9090 Columns 37 through 48 9.7596 99.4636 81.9344 157.8243 112.4600 94.4352 78.5318 109.4406 344.2906 12.3288 94.4934 106.0059 Columns 49 through 60 150.7689 155.1016 299.9237 299.6389 121.0485 68.2074 9.7211 100.8043 337.4185 280.4737 297.7058 283.4255 Columns 61 through 68 273.6078 266.0569 331.8528 140.6991 164.7559 113.5187 263.5157 110.5294 bandwidth = 263.5157 The matrix of Error is 1.0e+004 * Columns 1 through 12 0.2264 0.4125 2.0691 0.0331 0.0002 0.0279 0.0433 0.0211 0.2991 0.6475 0.0509 0.0459 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 308 Columns 13 through 24 0.4526 0.4624 0.6318 0.3921 0.1759 0.0451 0.0442 0.1399 0.1937 0.0433 0.3629 0.1302 Columns 25 through 36 0.0057 0.0012 0.5612 0.8049 0.9362 0.7362 0.0066 0.1432 0.7324 0.6111 0.6731 0.1487 Columns 37 through 48 0.0523 0.2523 0.2698 0.0367 0.0328 0.3069 0.1098 0.0450 0.6474 0.0473 0.5196 0.8569 Columns 49 through 60 0.2245 0.0367 0.6500 0.4379 0.0288 0.1767 0.6687 0.1408 0.6969 0.5965 0.5925 0.4422 Columns 61 through 69 0.4560 0.5661 0.3782 0.0712 0.0915 0.0331 0.0728 0.0332 0.0433 Cv = 3.1777e+003 The CROSS_VALIDATIOn factors 1.0e+003 * 2.0559 1.7628 1.7629 2.0098 2.5940 2.8404 3.1777 OpNei = 14 Beta = 1.0e+003 * 1.6125 0.0000 -0.0017 0.0003 THE PREDCTED BUS RUNNING TIME IS 721.66 SECONDS.Program complete Figure D4.2: An Example of the Matlab Outputs (Case 54-2200, R95) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 3 0 6 3 0 747 746 747 747 794.5 866.5 686.5 Observed 631 718.97 746 866.83 636.25 808.5 658.25 735.181 763.791 693.044557.671 677 681.5 667.602 Predict Observed Predict Observed 0 . 5 6 3 0 0 . 5 a(k+1) Predict 0.465350.34077 746.535 866.5 0.45499 0.47556 0 0 0 . 5 2167.1 6257.82 107.556 0.5 3002.16 0.38681 75.8485 0.03378 387.049 2865.24 0.48584 708.695 e(k+1) a(k+1) e(k+1) a(k+1) 0.5 911.25 0.5 0.5411 324.929 0.4589 0.54572 2470.8 0.45428 0.96622 g(k+i) 0.52444 0.51416 0.61319 9(k+1) 78.5 4896 1822.5 3 9 6 9 0 0 0 . 5 16592.4 0.53465 8871.1 9492.63 0.65923 4527.63 3 9 6 9 0 0 0 . 5 1573.44 0.5 786.722 0.5 3976.25 0.54501 5572.63 Var[data]n Var[data]n 729 529 5184 3 9 6 9 0 0 8010.25 A1 Var[data]n g(k+1) e(k+1) 2952.11 738.028 A1 860.444 215.111 0.5 625 2401 600.5 9025 396900 396900 A2 A1 36 121 25 396900 396900 9571.36 23613.4 3117.36 3 9 6 9 0 0 1573.44 1573.44 6293.78 738.028 738.028 215.111 215.111 Table D4.43: for 47-1200 D4.43: Table case Kalman Filter Table D4.42: D4.42: for 54-2200 Table case Kalman Filter Table D4.44: D4.44: for 48-1200 Table case Kalman Filter 630 396900 396900 396900 396900 0.5 0 771 9216 576 6 3 0 625 2550.25 5402.25 762 225 10920.3 8010.25 659.5656.5 2916 30.25 729 558.5 691.167 600.667 552.667 0 6 3 0 0 747 655 art1 Average A3 A2 553.5 672.5 642.5 671.833 art1 Average A3 A2 0 0 0 0 571 867 706.5 625 18360.3 25760.3 747 699 513 632 747 673 722 576 686,5 686.5 art2 art1 Average A3 art2 0 0 867 art3 art2 675.5 551.5 648 art3 1 686.5 686.5 19 662 561.5 746 18 605.5 20 789 537.5 1 7 1 7 0 19 18 513 19 746 2024 552.5 573.5 569.5 573.5 20 747 866.5 trip 8 : 3 3 1 7 8:39 8:43 8:59 24 681.5 8:36 8 : 3 3 8:39 8:43 8:59 8:59 24 8 : 3 3 8:36 8:43 8:39 8:36 time trip time time trip art3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 3 0 632 6 3 0 6 3 0 563 638 648.5 548.5 Observed Observed Observed 658 592 677.5 526 719.25 686 711.795 704.871735.215 632.5 607.173 663.749 726.084746.848 748 746.5 669.684 Predict Predict Predict 0.5 0.5 0.5 0.4707 0.34288 0.05693 0.47719 809.953 755 0 0.5 0 0.5 0 1394 0.5 123.66 555.245 0.09899 e(k+1) a(k+1) 3531.18 1300.85 e(k+1) a(k+1) 5465.71 574.952 0.09425 e(k+1) a(k+1) 86.5712 0.33978 0.5 0.5 0.5 0.5 0.5293 4992.59 g(k+i) 0.90101 0.57774 2040.99 0.42226 g(k+D 0.94307 g(k+D 2788 0.5 616.25 3 9 6 9 0 0 3 9 6 9 0 0 131.125 131.125 0.66022 1529.4444 0.76792 1174.49 0.23208 9432.3611 7062.3611 0.5 10454.444 0.52281 3532.6944 2601.6944 634.77778 0.90575 Var[data]n 3 9 6 9 0 0 93.4444 641.778 693.444 608.444 1521.1111 0.65712 999.546 2466.78 802.778 235.111 A1 Var[data]n A1 Var[data]n 6.25 182.25 396900 9248.03 592.111 A2 A1 850.694 2433.78 A2 A2 256 6.25 182.25 396900 396900 396900 396900 4876.69 608.444 A3 A3 Table D4.47: 21-1200 Table D4.47: Kalman case for Filter Table D4.46: Table case Kalman D4.46: Filter for 16-3000 Table D4.45: Table36-1200 Kalman D4.45: case Filter for 6 3 0 3 9 6 9 0 0 630 396900 396900 396900 6 3 0 663.5663.5 256 730.667 266.778 1002.78 696.667 1213.36 3990.03 Average 764.6667 8494.69 10370 712.3333 722.6667 2466.78 0 0 747 722.333 677 677 755 748 686 808.5 712.5 676 4900 9216 666.5 716.167 6214.69 866.5 801.833 4181.78 16727.1 4181.78 746.5 730.5 992.25 240.25art1 256 Average 0 0 0 687 666 746 633.5 725 art2 art1 art2 art1 Average A3 0 0 0 0 647.5 642.5 808.5 1 7 1 7 1819 686.5 642.5 747 673 19 795 18 731.5 1 7 24 747 699 746 24 20 866.5 672.5 20 647.5 666 18 19 673 747 20 672.5 866.5 24 699 trip art3 8 : 3 3 8:36 8:39 8:39 8:33 8:59 8:43 8:36 8:43 8:59 8 : 3 3 8:43 8:39 8:59 8:36 time time trip art3 art2 time trip art3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 3 0 613 546 6 3 0 630 747 773 746.5 698.5 Observed Observed 750.85 635.75 686 Predict Observed 554.982 553.5 576.104 644.668 547.5 Predict 723.821 643.5 717.949 741.636 747.5 746.766 746 0.5 570.75 0.5 658 0.5 0.1111 0.37442 0.49878 a(k+1) Predict 0.148290.46882 753.796 0 0 . 5 0 0 . 5 0 0.5 e(k+1) a(k+1) 2639.42 0.48401 e(k+1) e(k+1) a(k+1) 235.951 315.617 0.5 5672.13 0.5 40.625 9(k+1) 0.71642 2662.3 0.28358 0.51676 1757.8 0.48324 9(k+1) 0.55825 8778.21 0.44175 g(k+D 0.53118 1260.23 3 9 6 9 0 0 0 . 5 370.569 0.85171 8274.125 0.50122 4147.19 3716.1111 5115.2361 0.51599 Var[data]n 5776 11344.25 132.25 13086.125 0.62558 8186.43 396900 396900 0.5 2100.69 0.11111 380.27778 0.8889 338.029 1381.36 471.903 0.5 A1 34.0278 3401.57 A1 Var[data]n A1 Var[dataln 3782.25 373.778 53.7778 A2 1936 2809 81 2372.5 396900 396900 396900 396900 0.5 396900 396900 396900 386.778 890.028 3741.36367.361 3061.78 373.778 0.02778 Table D4.50: 39-1200 for D4.50: Kalman case Filter Table Table D4.48: for Kalman case Filter 19-1200 Table D4.48: Table D4.49: 45-1200 for Kalman case Filter Table D4.49: 6 3 0 738 741.667 592.1667 5525.44 1906.78 940.444 Average A3 A2 0 630 396900 396900 0 6 3 0 685 673.5 14400 11772.3 686 648.833 797 674.5 16512.3 36 15006.3 773747 772.833 562 607.8333 8220.44 2010.03 657 657.3333 753.5 697.667 22550 8898.78 3117.36 15724.403 641.5 624.5 132.25 30.25 289 81.25 art1 Average A3 art1 Average A3 A2 511.5 587.5 18906.3 561.5 0 0 0 0 638 563 0 0 619 641.5 613 619 677 1 7 1 7 18 18 19 546 680.5 19 680.5 797 747.5 2024 547.5 553.5 792 782 1 7 0 18 725 526 2024 792 782 753.5 685 19 666.5 548.5 20 24 698.5 trip art3 art2 trip art3 art2 trip art3 art2 art1 8 : 3 3 8 : 3 3 8:43 8:59 8:36 8:39 8:36 8:39 8:43 8:59 8 : 3 3 8:39 8:36 8:43 8:59 time time time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 3 0 6 3 0 6 3 0 7 5 2 755 694 747 688 614.5 Observed Observed 622.75 736.18 Predict Observed 672.307 559 711.767 682.748 763 651.622 563 Predict 713.198 746.5 0 . 5 0.5 679.25 0.49527 309.879 548 0.49984 0 0 . 5 0 0 . 5 1128.83 0.29291 176.278 695.877 0.41957 0.22222 0.5 683.75 2257.35 0.5 e(k+1) a(k+1) e(k+1) a(k+1) 459.637 0.26735 782.474 830.5 e(k+1) a(k+1) Predict 493.402 0.4548 0.5 0.5 0.5 0.99845 46.2893 0.00155 g(k+D 0.73265 0.58043 g(k+D 0.50016 0.61592 179.866 0.38408 782.709 g(k+D 2035.4 0.53645 1091.88 0.46355 3 9 6 9 0 0 3 9 6 9 0 0 0 . 5 0 59138.1 0.50473 29848.6 627.361 0.44444 352.444 484 905 0.5452 110.25 640.125 0.5 320.063 396900 396900 3 9 6 9 0 0 3 9 6 9 0 0 1201.78 1596.44 0.70709 1013.36 1198.9 1.77778 160.444 46.3611 831.361 2433.78 4514.69 0.5 205.444 592.111 292.028 A1 Var[data]n 900 289 23104 212060 160.444 113.778 14.6944 971.361 A2 A2 A1 Var[data]n A2 A1 Var[data]n 1521 95172.3 7598.03 1431.36 A3 A3 Table D4.51: Kalman for case 38-1200 Filter D4.51: Kalman Table Table D4.53: forcase Filter D4.53: 29-1200 Kalman Table Table D4.52: Kalman Filter for case D4.52: Kalman Table 64-2200 630 396900 396900 707 701.667 2844.44 348.444 778.167 283.361 736.167 0.44444 0.44444 699.833 220.028 2177.78 Average A3 Average 667 677 705.833 802.778 3268.03 61.5 522 728.5 718 380.25 792.5 737.5 672.5 685.167 78.0278 art1 0 0 630 396900 396900 674 689 683 747 679.5 677.5 666.833 544.444 art2 0 0 0 630 396900 396900 755 752 627 615.5 664.833 795 694 735.5 735.5 746.5 770.5 795 770.667 584.028 0.02778 830.5 677.5 763 1 7 0 0 0 1 7 18 19 643.5 19 18 1819 737.5 688 24 685 746.5 668 24 746 690 685 20 24 20 20 trip art3 art2 art1 Average trip art3 trip art3 art2 art1 8 : 3 3 8:36 8 : 3 3 1 7 0 8 : 3 3 8:59 8:36 8:39 8:59 8:43 8:59 8:39 8:36 8:39 8:43 8:43 time time time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 3 0 6 3 0 6 3 0 728 750 720 805.5 691.5 672.5 647.5 Observed Observed Observed 679 743 734.64 681 606.084 690.814 587.143 622.107666.629 810 605.448 695.818 0.5 0.5 a(k+1) Predict 0.34102 a(k+1) Predict 0.25977 0.42905 0.44857 0 0 0 . 5 1002.7 11299.4 3485.16 7498.89 0.5 609.75 700 e(k+1) a(k+1) Predict e(k+1) e(k+1) 2505.39 0.5 2721.73 0.24511 5304.89 0.5 569.5 0.5 0.5 0 0.5 0.56204 4931.02 0.43796 608.427 699 g(k+D 9(k+1) g(k+D 0.57095 4698.56 0.71668 2485.61 0.28332 0.55152 3884.92 0.44848 3 9 6 9 0 0 3 9 6 9 0 0 0 . 5 3 9 6 9 0 0 1354.57 0.74023 3468.24 5288.74 0.65898 7044.03 2811.25 0.57567 1618.36 0.42433 3605.44 0.75489 Var[data]n 4356 1694.69 5208.03 2100.69 5575.11 10609.8 396900 396900 396900 396900 1877.78 981.778 2417.36 3441.78 A2 A1 A2 A1 Var[data]n 5550.25 72.25 3 9 6 9 0 0 3 9 6 9 0 0 396900 396900 396900 12395.1 17600.4 455.111 14997.8 0.5 831.361 A3 5954.69 3885.44 6136.11 19787.1 5010.78 0.5 Table D4.56: Kalman FilterD4.56: for Table 8-3000 case Table D4.54: Kalman Filter for D4.54: 6-3000 Table case Table D4.55: Filter D4.55: for Kalman Table 56-2200 case 6 3 0 6 3 0 633 599.167 8160.11 587.333 580.667 3500.69 12958 2988.44 8229.36 639.333 8341.78 32640.4 7980.44 20491.1 0.55143 583.667 17777.8 593.833 1178.78 6032.11 1877.78 Average A3 A2 A1 Var[data]n Average 589.5 610.833 805.5 637.833 7714.69 6373.36 28112.1 550.5 0 0 743.5 571.5641.5 557 558.5 602.833 665.167 16986.8 560.111 11377.8 8773.44 671.5 558.5 699 0 0 0 0 717 525 509 571.5 557 672.5 600.333 521.5 694.5 526 499.5 641.5 art3 art2 art1 1 7 0 0 0 6 3 0 1 7 18 19 559.5 19 680 24 548 820 550 24 689.5 550 558 20 24 550 558 8 : 3 3 8 : 3 3 1 7 8:36 8:59 8 : 3 3 8:368:39 18 19 525 509 728 8:59 8:43 8:368:39 18 8:43 20 795.5 8:59 8:39 8:43 20 time trip art3 art2 art1 time trip time trip art3 art2 art1 Average A3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 3 0 781 747.5 778.5 Observed Observed 687.5 740 686.5 686.5 720.75 580 749.648 707.893 647.302 557 685.584 711.238 748.5 0 . 5 6 3 0 0 . 5 0 . 5 6 3 0 0.46682 0.48002 613.539 564 a(k+1) Predict 0.27974 0.48883 a(k+1) Predict Obbserved 0.38664 0.26892 690.155 746 2011.2 1477.12 e(k+1) a(k+1) Predict 449.868 0.5 e(k+1) 919.866 2399.25 236.535 0.5 8350.7 0.45471 e(k+1) 6856.25 0.5 3050.92 0.5 0 .5 0 0.5 0.8754 214.584 0.1246 723.761 542 9(k+1) 0.600640.53318 536.143 0.39936 705.163 687 g(k+D 0.51117 g(k+D 0.73108 2916.68 0.61336 0.54529 3 9 6 9 0 0 0 . 5 0 899.736 0.5 473.069 15314.1 13712.5 3989.57 4974.13 Var[data]n Var[data]n Var[data]n 380.25 245.125 6889 406.694 0.02778 5.44444 4693.61 396900 396900 2618.03 4624 396900 396900 1560.25 600.25 892.625 205.444 5088.44 3772.11 230.028 3441.78 2840.69 0.51998 15376 0.25 476.694 1034.69 A2 A1 3 9 6 9 0 0 330.028 1469.44 7338.78 4853.44 4533.78 22801 15252.3 A3 Table D4.57: 49-1200 D4.57: Kalman for Filter Table case Table D4.58: D4.58: 59-2200 Kalman Table for Filter case Table D4.59: 24-1200 D4.59: Kalman Table Filter for case 630 396900 396900 396900 396900 0.5 0 706.5 225 660 681.5 724.833 640.333 6944.44 Average 644.5 703.167 5451.36 811.5 811.667 469.444 681 691.5 art1 0 0 6 3 0 746 682 688 747.5 804.5 733.167 748.5 679.5 703 484 2070.25 552.25 1277.13 0.72026 763.5 746.5 766 484 6.25 805.5 art2 743 686.5 745 777 790 833.5 681 691.5 647.5 558 557 672.5 1 7 0 18 1 7 0 0 0 19 19 18 20 24 24 820 683 748 750.333 20 788 17 0 0 0 630 396900 396900 18 509 728 743 19 24 20 558.5 699 647.5 635 5852.25 4096 156.25 trip art3 art2 art1 Average A3 A2 A1 trip art3 art2 art1 Average A3 A2 A1 trip art3 8 : 3 3 8:36 8:39 8:39 8:43 8 : 3 3 8:59 8:59 8:43 8:36 8:59 8 : 3 3 8:36 8:39 8:43 time time time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 554 6 3 0 6 3 0 566 509 554.5 612.5 Observed Observed 605 676.25 Predict 641.419 599.062634.137 561.5 538.5 547.579 535.5 549.141 0 .5 0 .5 0.30224 0.47638 0.37651 0.44633 111.313 0.5 1229.65 0.2478 576.018 e(k+1) a(k+1) 2831.04 e(k+1) a(k+1) Predict g (k+ i) 0.55367 0.63448 2441.21 0.36552 g (k+ i) 3 9 6 9 0 0 0 . 5 0 1122.57 0.52362 587.798 426.944 0.69776 297.903 896.069 0.62349 558.691 1634.736 0.7522 6655.028 0.5 3327.51 0.5 Var[data]n 20164 5113.25 240.25 222.625 0.5 3 9 6 9 0 0 396900 396900 0.5 0 420.25 3 9 6 9 0 0 152.111 4646.69 7281.78 11165.4 3847.611 3906.25 A2 A1 A2 A1 Var[data]n 2 5 160.444 693.444 186.778 413.444 6320.25 A3 Table D4.61: Kalman Filter for case 20-1200 Table D4.50: Kalman Filter for case 25-1200 6 3 0 3 9 6 9 0 0 630 396900 396900 705.167732.167 367.361 1586.69 1877.78 205.444 584.028 2934.03 741.667 8433.36 4876.69 26136.1 632.167 3117.36 Average Average A3 0 686 672.333 557 662,667 580 722.5 707 art1 0 0 0 748 746.5 542 684 686.5 748.5746.5 681 678 811.5 644.5 564 art2 art1 art2 0 688 833.5 art3 1 7 1 7 0 19 686 18 19 18 712 20 772 20 763.5 24 685 646 24 683 8:59 8:43 8:36 8:39 8:43 8:59 8 : 3 3 8:36 8:39 8 : 3 3 time trip art3 time trip Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.02770318 0.16920071 0.016516245 0.003381038 0.176319444 0.050327321 0.024991023 0.165110132 0.129363697 0.008645533 0.095668135 0.152665382 0.162262658 0.153875339 0.905327979 0.032989276 0.206514745 0.137789685 0.179 0.0741 0.0498 0.0298 0.2027 0.1621 0.0788 0.0907 0.0027 0.0395 0.0002 0.0446 0.024 0.0028 0.1633 0.0412 0.0152 0.0134 0.0276 0.0744 0.0188 0.0253 700 0.0958 0.0608 544.85 0.0031 0.1578 557.11 846.95 543.08 0.0106 786.17 698.37 0.1046659.65 0.2476 0.083 0.132998138 663.32 0.0917 0.1597 900.06649.93 0.0675 0.02 0.0238 0.0005 643.64 0.0003 1005.69 0.0017 0.1604 0.346305221 713.2 647.3 549.14 550.32 672.31 707.89 735.21554.98 529.45 468.33 736.18 739.9 666.63 730.5 711.24 731.6 746.85 721.66 667.6 745.72 866.83 557.38 721.24 606.08 713.85 566.51 731.69 734.64562.93 746.15 630.23 746.24 760.47 548.75716.24 663.75696.85 717.47 658.26 746.17 70.72 721.39 0.0659 Table D4.62: Prediction Error ofthe Predictors 554 555.73 641.42 746 681 632 746 695.64 763.79 747 694 553.5 563.92 746.5 1200 1200 566 1200 747 1200 559 1200 748.5 1200 557 1200 746.5 1200 1200 1200 563 1200 1200 1200 2200 794.5 3000 3000 2200 805.5 2200 778.5 3000 720 2200 6 8 16 54 19 56 59 36 39 25 47 29 49 24 45 38 64 20 48 21 Note: d l, d2, d3: The absoluterelative (%) oerror the f Developed model, the Kalman Filter and theNaive models, respectively Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix D5 1. General This Appendix presents the database, the prediction results of the developed model, the Kalman Filter and the naive models applied for boarding passenger prediction. 2. Database for the Developed Model and Kalman Filter Model Table D5.1: Database of Boarding Passengers at Mackenzie Station (Route 95) f* R2 ■ 54 2200 3 5 794.5 (721.16) 681.5 58 2200 2 8 681.5 798 33 1200 7 5 571 537.5 42 1200 12 6 867 747 47 1200 9 7 747 (745.72) 866.5 62 2200 1 6 699 672.5 48 1200 11 10 746 (695.64) 866.5 36 1200 6 8 746.5 (730.63) 755 16 3000 9 9 632( 630.23) 632.5 52 2200 8 9 781.5 785 3 3000 12 3 803.5 647.5 2 3000 7 9 673.5 666 7 3000 3 6 698.5 677 21 1200 7 11 563( 548.75) 638 28 1200 8 11 562 657 19 1200 7 7 553.5 (563.92) 547.5 4 3000 5 16 782 792 11 3000 1 8 685 753.5 45 1200 10 8 747(716.24) 773 39 1200 7 10 746(696.85) 746.5 5 3000 6 6 690 770.5 55 2200 6 9 685 795 38 1200 9 9 746.5(746.24) 747 13 3000 7 9 668 792.5 64 2200 12 9 694(760.47) 830.5 14 3000 7 4 689 674 10 3000 4 0 672.5 661.5 29 1200 7 7 559(566.51) 548 35 1200 6 6 548 521.5 317 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 318 53 2200 11 7 820 694.5 17 1200 5 5 550 526 6 3000 6 6 720(739.39) 810 12 3000 9 6 689.5 795.5 18 1200 9 11 550 641.5 31 1200 7 5 558 558.5 56 2200 7 4 805.5 (721.24) 699 8 3000 8 6 681(731.69) 647.5 49 1200 6 4 748.5(730.5) 747.5 9 3000 7 8 679.5 804.5 59 2200 12 7 778.5(713.85) 781 41 1200 1 12 748 754.5 46 1200 10 4 747 766.5 57 2200 2 5 671 691.5 51 2200 9 10 694.5 777.5 1 3000 8 10 702 676.5 22 1200 8 7 556 540.5 40 1200 10 5 747.5 746.5 63 2200 14 10 798 796.5 50 1200 9 5 820 788 61 2200 9 10 683 763.5 43 1200 5 12 748 746.5 24 1200 8 13 557(562.93) 542 25 1200 7 2 566(557.38) 535.5 37 1200 1 5 746 869 65 2200 9 9 685 772 44 1200 6 5 646 746.5 60 2200 3 4 686 678 20 1200 8 10 554(555.73) 538.5 34 1200 10 8 553.5 657.5 26 1200 8 5 553 640 23 1200 6 7 569.5 548.5 30 1200 5 7 565.5 552.5 32 1200 12 3 572.5 569.5 27 1200 6 7 573.5 553.5 15 3000 11 9 655 673 66 1200 5 12 746.5 748.5 67 1200 2 10 755.5 866.5 68 1200 4 10 747 765.5 69 1200 10 11 756.5 866.5 70 1200 4 8 747 746 Note: 1. R1, R2, B 1, and B2 = Running time and Boarding passengers of trip 24 and trip 20 respectively. 2. Numbers in brackets are the predicted running times of the bus in question. These, along with B1, B2, and R2, are the elements of the pattern matrix defining the boarding passenger being predicted. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 319 Table D5.2: Database for Kalman Filter Predictor 54 2200 0.003776 0.007337 0.001477 0.006693 58 2200 0.0029347 0.010025 0.006042 0.001349 33 1200 0.0122592 0.009302 0.005343 0.013212 42 1200 0.0138408 0.008032 0.002681 0.002913 47 1200 0.0120482 0.008078 0.005355 0.00437 62 2200 0.0014306 0.008922 0.007429 0.004669 48 1200 0.0147453 0.011541 0.005355 0.006184 36 1200 0.0133958 0.010596 0.010695 0.004373 16 3000 0.0174051 0.014229 0.020046 0.003378 52 2200 0.0102367 0.011465 0.005827 0.006757 3 3000 0.0149347 0.004633 0.011321 0.006835 2 3000 0.0103935 0.013514 0.004367 0.012628 7 3000 0.0042949 0.008863 0.0015 0.015172 21 1200 0.0124334 0.017241 0.009116 0.007605 28 1200 0.0142349 0.016743 0.014248 0.009775 19 1200 0.0126468 0.012785 0.003663 0.019576 4 3000 0.0063939 0.020202 0.005878 0.009693 11 3000 0.0014599 0.010617 0.015056 0.006235 45 1200 0.0133869 0.010349 0.016054 0.008746 39 1200 0.0093834 0.013396 0.004662 0.001432 5 3000 0.0086957 0.007787 0.007358 0.008158 55 2200 0.0087591 0.011321 0.008856 0.013559 38 1200 0.0120563 0.012048 0.010485 0.00436 13 3000 0.010479 0.011356 0.002954 0.008236 64 2200 0.0172911 0.010837 0.005298 0.006649 14 3000 0.0101597 0.005935 0.004392 0 10 3000 0.005948 0 0.005997 0.006499 29 1200 0.0125224 0.012774 0.001776 0.003255 35 1200 0.0109489 0.011505 0.025022 0.006006 53 2200 0.0134146 0.010079 0.004468 0.006725 17 1200 0.0090909 0.009506 0.00545 0.013571 6 3000 0.0083333 0.007407 0.006667 0.008571 12 3000 0.0130529 0.007542 0.005882 0.006974 18 1200 0.0163636 0.017147 0.008749 0.009524 31 1200 0.0125448 0.008953 0.012567 0.009823 56 2200 0.0086903 0.005722 0.004461 0.010989 8 3000 0.0117474 0.009266 0.013015 0.009421 49 1200 0.008016 0.005351 0.004021 0.005827 9 3000 0.0103017 0.009944 0.005865 0.009396 59 2200 0.0154143 0.008963 0.008734 0.008108 41 1200 0.0013369 0.015905 0 0.004941 46 1200 0.0133869 0.005219 0.002679 0.011152 57 2200 0.0029806 0.007231 0.008785 0.002725 51 2200 0.012959 0.012862 0.004373 0.024506 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 320 1 3000 0.011396 0.014782 0.007391 0.006784 22 1200 0.0143885 0.012951 0.021661 0.003984 40 1200 0.0133779 0.006698 0.010724 0.004237 63 2200 0.0175439 0.012555 0.001481 0.009563 50 1200 0.0109756 0.006345 0.005148 0.006329 61 2200 0.0131772 0.013098 0 0.011998 43 1200 0.0066845 0.016075 0.001552 0.011091 24 1200 0.0071813 0.023985 0.001773 0.006897 25 1200 0.0123675 0.003735 0.00541 0.006531 37 1200 0.0013405 0.005754 0.009365 0.00424 65 2200 0.0131387 0.011658 0.004373 0.005618 44 1200 0.0092879 0.006698 0.005344 0.005827 60 2200 0.0043732 0.0059 0.007342 0.005536 20 1200 0.0108303 0.01857 0.003562 0.011788 34 1200 0.0180668 0.012167 0.007214 0.007775 26 1200 0.0144665 0.007813 0.010801 0.008056 23 1200 0.0105356 0.012762 0 0.00821 30 1200 0.0088417 0.01267 0.008795 0.012678 32 1200 0.0209607 0.005268 0.013323 0.003899 27 1200 0.0104621 0.012647 0.012693 0.001949 15 3000 0.0167939 0.013373 0.018519 0.004747 3. Prediction Results of the Developed Model The following table shows part of an output returned from Matlab source code for the sake of illustration. All predictions are tabulated in Table D5.23 The CROSSVALIDATlOn factors 11.6279 9.3710 9.6050 9.4747 9.5225 9.6709 9.7343 OpNei = 10 Columns 1 through 12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -0.8663 0.2075 -0.8663 -0.5083 -0.1504 -0.5083 0.9233 0.2075 0.5654 0.5654 -1.5821 0.5654 1.2755 -0.0305 -1.3076 2.1134 0.7265 0.1718 0.7150 0.7207 -0.6026 1.1252 1.3795 -0.1229 -0.2401 0.9386 -1.6971 0.4226 1.6317 -0.3312 1.6317 0.5036 -0.7359 0.8071 -0.5841 -0.3970 Columns 13 through 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 321 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -0.5083 1.2812 1.2812 -0.1504 3.0708 0.2075 0.2075 0.9233 -0.5083 0.5654 0.5654 0.5654 0.1660 -1.4000 -1.4116 -1.5098 1.1310 0.0100 0.7265 0.7150 0.0678 0.0100 0.7207 -0.1865 -0.2857 -0.6803 -0.4880 -1.5959 0.8779 0.4884 0.6857 0.4176 0.6604 0.9083 0.4226 0.8830 Columns 25 through 36 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.5654 -1.2242 -2.6559 -0.1504 -0.5083 -0.1504 -0.8663 -0.5083 -0.5083 1.2812 -0.8663 -1.2242 0.1140 0.0562 -0.1345 -1.4462 -1.5734 1.5702 •-1.5503 0.4145 0.0620 -1.5503 -1.4578 1.4026 1.2675 -0.3160 -0.4425 -1.5909 -1.8590 -0.1086 -1.8135 1.0601 0.9133 -0.6448 -1.4846 -0.0631 Columns 37 through 48 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -0.5083 -1.2242 0.2075 -0.1504 1.6392 -1.2242 -0.8663 0.9233 0.9233 -0.1504 -0.8663 0.9233 -0.0363 0.7438 -0.0536 1.0906 0.7381 0.7265 -0.1518 0.1198 0.2064 -1.4809 0.7323 1.3159 -0.5841 0.4277 1.0044 0.7666 0.4985 0.6199 -0.1389 0.7312 -0.2907 -1.6668 0.4176 0.9235 Columns 49 through 60 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -0.8663 0.9233 1.6392 -1.9400 -0.8663 0.5654 -0.8663 -1.2242 0.9233 0.2075 -0.8663 -0.1504 1.5702 -0.0131 0.7381 -1.3653 0.7150 0.0100 ■•0.4408 0.0215 -1.5040 -1.5098 - 1.5156 -1.3249 0.8375 0.5896 0.4176 -1.7174 1.6570 0.6756 0.4176 -0.2755 -1.6870 -0.4830 -0.6600 -1.5858 Columns 61 through 69 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -0.1504 -1.5821 -0.1504 0.5654 1.6392 0.9233 0.9233 1.2812 0.2075 -1.3711 -1.2902 -1.2787 -0.3367 0.7207 0.8247 0.7265 0.8363 0.7265 -1.5454 -1.3733 -1.5352 -0.3261 0.4378 1.6317 0.6098 1.6317 0.4125 boarder = 8 THE REAL-TIME PREDCTED # of PASS OF THE BUS IS 8.00 PAS.Program complete.HAVE A NICE DAY Figure D5.1: An Example of the Matlab outputs for Boarding Passenger Prediction (Case 24-1200) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 322 5 8 10 Boarders 0.025 0.025 0.00618 0.00808 0.00734 0.00669 0.00437 Observed Observed Boarders Observed Boarders 0.013 0.004 0.006 0.015 0.011 0.012 Predict Predict 0.00841 0.01154 0.01483 0.01312 0.00148 0.01487 0.00277 0.00536 0.5 0.50.5 0.025 0.5 0.054 a(k+1) a(k+1) 0.14489 0.22198 0.00744 0.45663 0 0 0 0.5 07 08 07 05 0.43458 07 06 0.47178 0.00776 06 05 0.5 0.01396 06 0.49885 07 0.40523 0.00569 0.00536 05 0.49684 06 1.3E- 1.6E- 1.9E- 1.9E- 1.IE- 7.8E- 7.2E- 3.3E- 8.3E- 6.6E- 3.2E- 4. IE- 4. e(k+1) e(k+1) 0.5 0.5 0.5 fl(k+1) a(k+i) g(k+D e(k+1) a(k+1) Predict 0.56542 0.77802 0.52822 0.54337 0.85511 0.50316 0.0006 1.3E-06 1.6E-05 0.50115 3.7E-05 0.5 3.5E-06 8.4E-08 6.5E-07 4.3E-07 Var[data]n 06 07 2.8E-05 06 06 1.6E-05 06 05 05 0707 6.9E-07 0.59477 07 05 2.6E-05 1.7E- 1.3E- 1.5E- 1.2E- 5.2E- 8.7E- 8.5E- 3.3E- 5.9E- 8.5E- 4.7E- 0.0006 06 05 08 06 05 06 06 06 08 07 06 07 4E-06 1.1E-06 0.946 1.3E- 1.5E- 4.9E- 5.5E- 2.5E- 7.1 E- E- 7.1 6.7E- 3.IE- 3.3E- 4.6E- 8.7E- 4.3E- 07 08 06 05 07 05 06 05 08 05 06 1.3E- 1.IE- 1.8E- 2.7E- 9.7E- 2.4E- 8.2E- 2.3E- 2.2E- 5.4E- 4.5E- 0.0006 0.0006 0.0006 0.0006 0.5 0.0006 0.0006 Table D5.4: Kalman forFilter 47-1200 test case Kalman Table D5.4: Table D5.3: for testFilter 54-2200 Tablecase Kalman D5.3: Table D5.5: 48-1200 Filter for test case TableD5.5: Kalman 0.025 0.0006 0.0006 0.0006 0.0006 0.5 0.01043 0.00834 Average A3 A2 A1 Var[data]n Average A3 A2 A1 Var[data]n 0 0.025 0 0.025 art1 art1 0.01337 0.00475 0.00353 0.00892 0.00467 0.00398 0.01384 0.00968 0.01679 0.01607 0.00291 0.00582 2E-05 0 art2 art2 art2 art1 Average A3 A2 A1 0.0093 0.00803 0.00912 0.01265 0.01269 0.01852 0.01485 0.00437 0 0.00527 0.00803 0.00808 0.00293 0.01226 0.00604 0.00534 0.00268 0.00469 0.01384 0.01205 0.00143 0.00911 0.00604 0.00743 0.00536 0.00628 18 0.003919 0.00195 0.01332 18 0.00291 19 20 24 0.02096 0.01046 20 0.01003 8:43 20 8:36 8:39 8:33 17 0 0 0 8:33 178:36 0 0 8:59 8:33 17 8:43 8:368:39 18 0.00135 0.01321 8:59 24 8:398:43 19 8:59 24 time trip art3 time trip art3 time trip art3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 323 6 9 10 Boarders 0.025 0.025 0.025 0.0107 0.00338 0.00761 0.00912 0.00437 Observed Observed Boarders 0.014 0.017 0.013 0.013 Predict Observed Boarders 0.01559 0.00723 0.02005 0.00897 0.01724 0.01121 0.0106 0.5 a(k+1) Predict 0.47365 0.46774 0.0049 0.49489 0.36957 0.01535 0.01423 0.42539 0.0041 0.38542 0.009 0.012 0 0.5 0 0.5 05 0.49236 0707 0.5 05 0706 0.5 0.01469 07 0.39565 06 06 05 06 0.505 0.02009 0.4033 1.6E- 1.6E- 1.5E- 1.2E- 1.8E- 6.4E- 7.3E- 1.2E- 3.5E- 2.4E- 5.8E- 9.5E- e(k+1) e(k+1) a(k+1) Predict e(k+1) a(k+1) 0.5 0.5 g(k+i) g(k+i) g(k+D 0.50764 0.60435 0.52635 0.57461 7E-07 2E-05 0.61458 2E-05 0.5967 0.0006 0.0006 0.5 0 0.0006 0.5 1.2E-05 0.5 4.7E-05 Var[data]n 06 3.1E-06 05 07 1.2E-06 0.53226 06 3.3E-07 0.5 07 06 1.2E-06 05 3.5E-05 0.50511 06 2.3E-06 0.63043 05 1.7E-05 06 05 08 A1 Var[data]n 1.2E- 1.3E- 1.3E- 1.8E- 1.IE- 8.2E- 4.8E- 4.1 E- 4.1 2.8E- 4.9E- 2.5E- 2.2E- 06 05 06 07 05 06 07 07 06 05 06 05 A2 1.5E- 1.2E- 1.2E- 1.9E- 3.5E- 1.9E- 6.1 E- E- 6.1 6.4E- 2.4E- 2.7E- 2.5E- 2.IE- 0.0006 0.0006 0.0006 0.0006 07 07 07 06 05 05 05 07 06 05 05 1.7E- 1.6E- 1.5E- 5.IE- 2. IE- 2. 7.1 E- E- 7.1 1.9E- 3.IE- 4.8E- 4.4E- 2.2E- 7E-06 0.0006 0.0006 0.0006 e D5.8: D5.8: 21-1200 e test case Kalman Filter for Table D5.6: 36-1200 testD5.6: for Kalman Filter Table case Tab Table D5.7: for D5.7: testTable Kalman Filter case 16-3000 0.025 0.0006 0.00605 0.00941 0.00508 0.01102 0.01155 Average A3 A2 A1 Var[data]n Average A3 0 0.025 0 0 0.025 0.0006 art1 0.0107 0.00783 0.0126 0.00618 0.00508 0.00437 0.01154 0.00951 0.00429 0.00534 0.01517 0 0 art2 art2 art1 Average A3 A2 A1 0.00469 0.00892 0.00618 0.01154 0.01475 0.0134 0.00986 0.00437 0.0015 0.00573 0.01029 0.01354 0.00886 0.00901 0 0 0 0 art3 art3 art2 art1 art3 0.00437 0.00467 0.00892 0.01132 0.00685 0.01263 0.00463 17 17 18 17 19 0.00536 0.00743 0.00536 18 19 20 20 20 0.00808 trip trip trip 8:33 8:39 8:33 8:36 8:39 19 0.00743 0.00536 8:43 8:59 24 0.00143 8:39 8:33 8:36 8:43 8:59 24 0.01205 0.00143 0.01475 8:59 24 0.00143 8:36 18 8:43 time time time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 324 8 4 10 Boarders Boarders Boarders 0.025 0.025 0.025 0.0134 0.00761 0.01279 0.01035 0.00912 Observed Observed 0.014 0.013 Predict Predict Observed 0.01138 0.00891 0.01687 0.00761 0.01559 0.00761 0.01196 0.00912 0.5 0.5 0.01739 0.5 0.5 0.5 a(k+1) 0.3709 0.01391 0.43247 0.49821 0.006 0.013 0.46774 0.00641 0.00912 0.33744 0.013 0.009 0 05 07 0.1886805 06 05 0.4848905 0.4017 0.00985 06 05 0.38638 0707 0.5 05 05 1.3E- 1.3E- 1.2E- 1.6E- 1.IE- 1.6E- 1.5E- 1.3E- 1.5E- 7.3E- 1.IE- 6.4E- e(k+1) a(k+1) e(k+1) a(k+1) Predict 0.5 g(k+t) g(k+i) 0.56753 0.61362 0.53226 0.66256 0.0006 0.5 0.0006 0.5 0 1.9E-07 0.81132 2.6E-05 0.5983 3.3E-07 2.2E-05 0.50179 2.5E-05 0.51511 Varfdatajn Var[data]n 07 3.1E-06 0.5 0506 2.3E-05 1.9E-05 0.6291 05 05 2E-05 06 1.5E-05 0.5 06 08 06 05 07 1.2E-06 A1 A1 Var[data]n e(k+1) A1 1.5E- 1.2E- 1.2E- 6. IE- 6. 1.IE- 1.4E- 3.6E- 8.6E- 5.9E- 2.7E- 4.8E- 0.0006 0.0006 0.5 0 07 06 06 06 05 07 08 05 06 05 4E-05 1.6E-05 06 1.IE- 1.5E- 1.9E- 8.8E- 6.9E- 9.6E- 4.5E- 3.5E- 7.4E- 9.1 E- 9.1 3.2E- 0.0006 0.0006 0.0006 05 05 06 4E-06 05 05 05 06 05 07 07 07 07 1.9E- 4.6E- 2.9E- 3.6E- 5.1 E- E- 5.1 2.9E- 2.2E- 4.2E- 8.8E- 4.2E- 4.7E- 4.8E- 0.0006 0.0006 0.0006 0.0006 0.0006 Table D5.9: for case Table D5.9: test Kalman Filter 19-1200 Table D5.10: 45-1200 case Table D5.10: test for Filter Kalman Table D5.ll: 39-1200 for test Filter case Kalman 0.025 0.025 0.025 0.01085 0.01372 0.01032 0.00439 0.00708 0.00605 Average A3 A2 0 0 0.01035 0.00978 0.00146 0.00536 0 0 art2 art1 Average A3 A2 art2 art1 Average A3 A2 art2 art1 0.01724 0.01674 0.01428 0.00892 0.00886 0.00862 0.00624 0.00875 0.00822 0.01062 0 0 0 0 art3 art3 art3 0.0202 0.0015 0.00912 0.01425 0.00829 0.01517 0.00761 0.00588 0.01506 0.01605 0.01233 0.00437 0.00469 0.00618 0.00508 17 0 18 18 20 24 0.00639 0.00146 0.01339 trip trip trip 8:39 19 8:43 208:59 0.00886 24 0.00429 0.01243 0.01423 8:36 8:33 17 8:36 8:33 8:33 17 8:36 18 0.00969 8:43 8:59 24 0.00143 0.01029 8:39 19 8:59 8:43 20 0.00808 8:39 19 0.00536 0.00743 time time time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 325 5 8 7 Boarders 0.025 0.025 0.025 0.017 0.0108 0.00912 0.01277 0.00178 0.00761 Observed Boarders Observed Boarders 0.009 0.013 0.011 Predict Predict Observed Predict 0.00641 0.00912 0.00891 0.01928 0.00753 0.01559 0.00326 0.5 0.5 0.5 a(k+1) 0.43334 0.01 0.013 0.18868 0.44317 0.01034 0.01204 0 0.5 0 0.5 0 06 0.19079 0.00862 07 0.46774 0.00437 07 0.18868 06 06 07 0.5 0.01559 0.00761 06 06 0.498 07 07 07 0.46774 05 0.49823 1.6E- 1.IE- 1.6E- 1.6E- 1.6E- 9.9E- 2.5E- 5.4E- 9.8E- 6.4E- 6.4E- 9.9E- e(k+1) a(k+1) e(k+1) a(k+1) e(k+1) 0.5 0.5 0.5 0.5 fl(k+1) g(k+i) g(k+i) 0.81132 0.56666 2E-05 0.5 0.0006 0.0006 0.0006 1.9E-05 0.502 1.9E-07 0.81132 1.2E-06 0.53226 1.9E-07 9.7E-06 0.55683 3.3E-07 Var[data]n Var[data]n Var[data]n 06 3.3E-07 0.5 05 06 3.1E-06 0.80921 08 06 1.7E-05 05 08 07 06 08 06 2.2E-05 0.50177 07 1.2E-06 0.53226 A1 A1 1.2E- 1.2E- 3.9E- 8.2E- 5.9E- 9.5E- 3.4E- 5.IE- 2.7E- 5.9E- 4.8E- 4.8E- 0.0006 07 06 06 08 07 06 05 07 08 07 06 A2 A1 A2 1.9E- 1.IE- 1.9E- 1.5E- 1.5E- 2.2E- 8.4E- 9.5E- 9. IE- 9. 9.1 E- E- 9.1 8.4E- 2E-07 05 06 05 05 07 07 07 07 06 07 07 05 A3 A3 A2 8.8E- 5.IE- 3.9E- 5.9E- 3.8E- 2.9E- 3.6E- 2.7E- 2.9E- 4.8E- 5.IE- 4.8E- 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 Table D5.12: 38-1200 test for Kalman Table Filter case D5.12: Table D5.13: for test 64-2200 D5.13: Kalman Filter Table case for 29-1200 test D5.14: case Kalman Filter Table 0.025 0.025 0.0074 0.00862 0.00605 0.00508 0 0 0.025 0 art1 Average art1 Average A3 0.00886 0.00689 0.00886 0.00618 0.00508 0.01356 0.00772 0.00595 0.01113 0.00886 0.00862 0.00618 0.01048 0 art2 art1 Average art2 0.00469 0.00816 0.00743 0.00536 0.00605 0.00892 0 0 0 0 0 art3 art3 0.014 0.00779 0.01132 0.01103 0.00536 0.00437 0.00143 0.00437 0.00469 0.00536 0.00743 0.00536 18 19 0.00446 0.00736 19 19 18 20 20 0.00808 8:33 17 8:33 17 8:368:39 18 8:59 24 0.00134 0.00939 0.00876 0.0065 8:59 24 0.01729 0.01016 8:43 20 0.00808 0.00892 8:39 8:43 8:33 17 8:36 8:39 8:43 8:36 time trip art3 art2 8:59 24 0.00143 0.01029 time trip time trip Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 326 6 5 7 Boarders 0.025 0.025 0.012 0.009 0.008 0.0074 0.00572 0.00923 0.00942 0.00446 Observed Boarders Observed Boarders Observed Predict Predict 0.00692 0.0013 0.00753 0.5 0.01799 a(k+1) 0.46774 0.00437 0.00178 0.27209 0.009 0.49663 0.49742 0.008 0.45976 0.00369 0 0.5 0.025 0 0.5 07 06 0.16845 0.011 05 0.47302 0.00683 0706 0.5 0.45753 0.01741 0.01185 0.01099 07 0.507 0.01559 0.18868 06 0.00326 06 07 05 1.IE- 1.6E- 1.6E- 1.2E- 1.3E- 6.4E- 7.6E- 5.3E- 2.7E- 9.5E- 2.8E- 4E-06 e(k+1) a(k+1) Predict e(k+1) a(k+1) 0.5 0.5 g(k+D 9(k+1) g(k+U e(k+1) 0.50337 0.54247 0.53226 0.54024 0.0006 0.0006 0.5 0 0.5 0.0006 0.5 1.5E-05 0.50258 3.3E-07 0.5 7.3E-06 0.72791 2.4E-05 0.52698 2.3E-05 Var[data]n Var[data]n Var[data]n 07 1.2E-06 05 7.9E-06 06 3.3E-06 0.83155 06 05 5.1E-06 08 1.9E-07 0.81132 06 1.9E-06 0.5 06 06 05 07 2.1E-07 05 1.2E- 1.2E- 1.IE- 1.5E- 7.7E- 1.7E- 5.9E- 2.IE- 5.1 E- E- 5.1 4.8E- 9.5E- 2.4E- 0.0006 10 06 07 06 05 06 05 08 07 08 06 1.9E- 1.5E- 1.4E- 1.6E- 9.1 E- 9.1 8.4E- 2.7E- 5.6E- 3.5E- 5.6E- 4. IE- 4. 1E-07 0.0006 0.0006 0.0006 0.0006 0.0006 07 05 07 07 06 07 05 05 07 08 05 A3 A2 A1 A3 A2 A1 1.5E- 1.3E- 2.9E- 5. IE- 5. 3.5E- 4.3E- 8.7E- 4.8E- 3.2E- 2.6E- 2.5E- 0.0006 0.0006 Table D5.15: testfor Filter Kalman D5.15: case Table 6-3000 Table D5.17: for test Kalman Filter D5.17: case Table 8-3000 Table D5.16: testFilter for D5.16: Kalman case 56-2200 Table 0.025 0.025 0.025 0.0006 0.00605 0.01218 0.01399 0.00877 0.01253 0.01011 Average Average 0 0 0 art1 art1 art1 Average A3 A2 A1 0.00536 0.00982 0.00869 0.01099 0.01257 0.00907 1E-05 0 0 art2 art2 0.01715 0.00895 0.01121 0.00952 0.01254 0.00982 0 0 0 0 art3 0.00536 0.00743 0.00754 0.00808 0.00892 0.00886 0.00862 0.00437 0.00469 0.00618 0.00508 0.00697 0.00875 0.01257 0.00446 0.00859 0.00952 0.01715 0.00895 0.00572 0.01061 19 18 19 18 20 20 trip art3 8:33 17 8:39 8:59 24 0.01636 8:39 8:33 17 8:43 20 8:39 19 0.00588 0.00875 8:43 8:36 8:59 24 0.01729 0.01016 0.00909 8:33 17 8:36 18 8:36 8:43 8:59 24 0.01305 0.01636 0.01255 time time trip time trip art3 art2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 327 6 7 7 Boarders 0.0069 0.0013 0.02399 0.00942 0.00177 0.00942 Observed Boarders 0.01 0.015 0.009 0.009 0.011 0.007 0.0172 0.0172 Predict Observed Predict 0.00757 0.0013 0.01805 0.00785 0.00535 0.5 0.025 a(k+1) 0.47958 0.22859 0.00797 0.00896 0.24205 0.38089 0.00356 0.49562 0.00763 0 0.5 0.025 0 0.5 0.025 0605 0.5 06 06 0.33173 06 0.5 06 0.33374 06 0.44186 0.00976 06 07 0.5 06 06 0.25025 06 1.IE- 1.8E- 2.3E- 3.3E- 3.5E- 3.4E- 9.8E- 3.6E- 2.3E- 4.2E- 4.2E- 8.9E- e(k+1) a(k+1) Predict Observed Boarders e(k+1) a(k+1) e(k+1) 0.5 0 0,5 o(k+D g(k+o 0.52042 0.66827 0.61911 0.50438 0.55814 0.0006 0.0006 0.5 0.0006 0.5 1.9E-05 2.1E-05 4.6E-06 0.75795 6.9E-07 0.5 4.6E-06 0.77141 6.8E-06 4.4E-06 0.74975 Var[data]n ter for test for 24-1200 ter case 08 3.5E-06 06 06 07 06 3.6E-06 06 08 3.5E-06 0.66626 0607 8.4E-06 0.5 06 0505 1.6E-05 A1 Var[data]n g(k+1) A1 Var[data]n A1 1.4E- 1.8E- 1.3E- 1.7E- 7.8E- 3.6E- 3.IE- 7.8E- 3.1 E- E- 3.1 5.1E- 3. IE- 3. 2.9E- 0.0006 06 06 05 06 07 05 06 06 1.IE- 1.3E- 1.6E- 5.7E- 8.4E- 4.9E- 2.6E- 4.8E- 4E-06 0.0006 0.0006 0.0006 06 05 05 06 8E-06 06 05 06 8E-06 07 06 A3 A2 1.5E- 1.2E- 1.IE- 1.2E- 1.2E- 2.9E- 8.6E- 2.8E- 4.9E- 3E-05 3E-06 0.0006 Table D5.19: 59-2200 D5.19: Kalman case test Filter for Table Table D5.18: 49-1200 D5.18: test case Kalman for Filter Table Table D5.2(): Kalman Fill D5.2(): Table 0.01002 3E-06 4E-06 0.00819 0.01184 0.01028 0.00222 Average 0 0.025 0.0006 0 0.025 0.0006 0.0006 0.0006 0.0094 0.00994 0.0103 0.01002 0.00994 0.01608 0.00994 0.00819 0 0 0 0.025 art2 art1 Average A3 A2 art2 art1 Average A3 A2 0.012 0.01109 0.00981 0.0131 0.00535 0.00402 0.00587 0.00763 0.00535 0 0 art3 art2 art1 art3 0.00927 0.00942 0.00583 0.0094 0.00821 0.00927 0.00635 0.00515 0 0.0015 17 0 17 18 18 0.00633 20 20 8:33 8:36 8:59 24 0.01175 0.00802 0.0103 8:39 19 0.01302 8:33 178:36 18 0 0.010998:43 0.00942 8:43 20 8:43 8:59 24 0.01098 0.01318 0.00668 8:59 24 0.01175 0.00802 8:33 8:36 8:39 19 time trip 8:39 19 0.01302 0.0044 0.00587 0.00776 time trip time trip art3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced Table D5.21: Kalman Filter for test case 25-1200 ev I I B v z £ "3 < 2 CO + + r tr to (0 o s I a(k+1) | Observed | Boarders | 90000 0 9000 00 0 9000 I I I I I time I trip I I Average I I Var[data]n I Predict | o o O T— N» o 8:33 0.025 | I 0.0006 | t f -a 0.5 | 0.025 | 90 0 6900 90 69000 2.7E- SO 00 8:36 0,012 0.01191 0.01027 3E-06 06 2.8E-06 0.5 0.5 0.01595 1.2E- 4.4E- 5.3E- o o o o h- h- o CM IO o CO o CO o to o o> o 8:39 0.00155 0.00177 0.00111 06 2E-07 07 7.1E-07 0.74969 07 t'9 a - ^ o ^ WlO CM 2.7E- 3.9E- 90 d ^r oo o> T o d CO d o to 00 o CM o 8:43 — 0.02399 0.01772 06 05 1.2E-05 0.51086 0.01312 0.00374 1.7E- 5.4E- 3.4E- 6.9E- CO o o 05 o CO o CO CO CD CO o d O O 05 o o CO CM t CO 8:59 T 0.00668 0.00718 05 06 06 1.1E-05 0.60632 06 0.012 fS O a ® F-H ® fS s H O r Q 13 -4-4 a u u CO 0> « to • e • s u c 4» i £ 3 < < CO + + CO CM + r r t ra 1 time trip 1 re re re Average Var[data]n Predict Observed Boarders | o s 90000 00 0 9000 0 9000 0 9000 9ZOO 9ZOO \ o O o o 8:33 N. 0.5 0.025 3.3E- 3.9E- 1.9E- SO 8:36 CO 0.00562 0.00587 0.00554 0.00568 09 08 08 2.1E-08 1E-08 0.5 0.01527 0.0069 1.7E- 1.2E- 2.7E- 4.6E- d h- b- O O d o o c r-- 05 o 8:39 0.00534 0.00734 L 0.00569 06 07 06 9.2E-07 0.50283 07 0.49717 0.00712 6S000 1.3E- 1.9E- 4.8E- 3.8E- UJ o CO CO ) 0 d o o CO O 00 CO CM O 0.01166 0.0067 05 06 06 0.51528 06 0.48472 0.0039 0.01857 1.8E- 1.3E- 2. IE- 5.2E- o o o d 5; CO CO o O) 111 o CO CM GO o CO 05 CO d o I 8:59 CM O' 0.01314 0.00929 0.00437 05 07 05 0.58762 06 329 Table D 5.23: Prediction Error of the 3 Predictors 54 2200 3 5 8 6 0.667 1.66667 1 47 1200 9 7 10 14 0.222 0.11111 0.555556 48 1200 11 7 5 2 0.364 0.54545 0.818182 36 1200 6 7 9 9 0.167 0.5 0.5 16 3000 9 7 10 11 0.222 0.11111 0.222222 21 1200 7 9 6 6 0.286 0.14286 0.142857 19 1200 7 6 8 5 0.143 0.14286 0.285714 45 1200 10 6 4 1 0.4 0.6 0.9 39 1200 7 7 10 10 0 0.42857 0.428571 38 1200 9 7 8 6 0.222 0.11111 0.333333 64 2200 12 7 8 7 0.417 0.33333 0.416667 29 1200 7 6 5 4 0.143 0.28571 0.428571 6 3000 6 8 5 6 0.333 0.16667 0 56 2200 7 5 7 5 0.286 0 0.285714 8 3000 8 4 6 11 0.5 0.25 0.375 49 1200 6 7 7 5 0.167 0.16667 0.166667 59 2200 12 7 7 6 0.417 0.41667 0.5 24 1200 8 8 7 5 0 0.125 0.375 25 1200 7 8 3 1 0.143 0.57143 0.857143 20 1200 8 7 7 8 0.125 0.125 0 Note: dl.d2, d3= Absolute Relative Error the for Developed Model, KF model, and the naive model, respectively Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix D6 1. General This Appendix presents the database, the prediction results of the developed model, the Kalman Filter, and the Naive model applied for alighting passenger prediction. 2. Database for the Developed Model and Kalman Filter Model Table D6.1: Database of Alighting Passengers at Mackenzie Station (Route 95) 54 2200 23 27 23 58 2200 21 26 23 33 1200 25 28 21 42 1200 12 13 13 47 1200 11 14 13 62 2200 21 23 27 48 1200 14 17 13 36 1200 13 14 10 16 3000 17 17 16 52 2200 28 28 23 3 3000 17 17 17 2 3000 17 16 17 7 3000 17 16 17 21 1200 22 33 25 28 1200 22 31 25 19 1200 23 33 23 4 3000 17 17 15 11 3000 17 17 17 45 1200 14 16 12 39 1200 13 14 11 5 3000 17 17 15 55 2200 23 31 20 38 1200 12 14 12 13 3000 17 17 17 64 2200 22 21 22 14 3000 17 17 16 330 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 331 10 3000 17 17 16 29 1200 25 28 20 35 1200 25 27 22 53 2200 28 21 26 17 1200 25 35 22 6 3000 17 17 15 12 3000 17 17 17 18 1200 29 27 23 31 1200 25 28 28 56 2200 22 22 20 8 3000 17 17 17 49 1200 11 14 13 9 3000 17 17 17 59 2200 27 29 25 41 1200 13 17 9 46 1200 12 17 12 57 2200 24 24 19 51 2200 26 29 22 1 3000 17 17 16 22 1200 27 26 23 40 1200 12 12 13 63 2200 25 25 22 50 1200 12 14 11 61 2200 20 30 22 43 1200 12 14 11 24 1200 23 29 26 25 1200 26 29 25 37 1200 15 16 13 65 2200 20 22 21 44 1200 11 14 13 60 2200 25 23 23 20 1200 26 26 23 34 1200 27 26 27 26 1200 24 29 26 23 1200 25 26 19 30 1200 21 27 22 32 1200 20 22 23 27 1200 19 26 23 15 3000 17 17 16 Note: A l, A2, and A3 = Alighting Passengers of trip 24th, trip 20*, and trip 19* , respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 332 3. Prediction Results of the Developed Model The following table shows part of an output returned from Matlab source code for the sake of illustration. All predictions are tabulated in Table D6.22 The CROSS_VALIDATIOn factors NaN 4.2983 4.0432 4.2601 5.0041 6.0432 7.1430 8.0114 OpNei = 15 Columns 31 through 45 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2.1420 -0.7306 -0.7306 0.8653 1.0249 0.0673 -0.7306 -1.2094 -0.7306 1.1845 -0.7306 -0.7306 0.3865 1.1845 -0.7306 0.6395 -0.7446 -0.3491 0.8372 1.8258 0.2441 -0.3491 -1.1400 -0.3491 1.2327 -1.9309 -1.3377 0.0463 0.6395 -0.5468 Columns 46 through 60 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.7057 -1.5286 0.5461 -1.2094 1.3440 -1.2094 1.1845 1.1845 -0.8902 0.0673 -1.2094 0.2269 0.7057 1.1845 0.7057 0.8372 -1.1400 0.6395 -1.5354 0.6395 -1.5354 1.4304 1.2327 -1.1400 0.4418 -1.1400 0.8372 1.6281 1.4304 0.0463 Columns 61 through 64 1.0000 1.0000 1.0000 1.0000 0.8653 0.0673 0.7057 -0.7306 0.6395 0.8372 0.8372 -0.5468 20.6431 2.8937 2.0531 deboarder = 25 THE REAL-TIME PREDCTED # of PASS OF THE BUS IS 25.00 PAS.Program complete.HAVE A NICE DAY Figure D6.1: An Example of Matlab output for de-boarding Prediction (Case 20-1200) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 2 17 14 2 2 2 2 13 11 27 23 23 Observed 14.5 13 20 11.5 Predict Observed 20.7189 Predict Observed 26.6364 14 0 . 5 0.5 9 0.5 a(k+1) a(k+1) Predict 0.40380.1256 19.4228 18 0.36077 20 0.09091 0 0 .5 0 0 . 5 0 0 . 5 24.5 0.5 e(k+1) e(k+1) 0.82583 6.56991 e(k+1) a(k+1) 1.10204 0.44898 0.45455 0.5 0 . 5 0 0 .5 0.5 g(k+i) 0 . 5 0.8744 g(k+i) 0.77189 7.93328 0.22811 g(k+i) 0.55102 9 0.5 4.5 4 0 0 49 9 0.5 4.5 2 4 0 0 0 . 5 9.444440.94444 0.5962 5.63077 0.5 0.90909 4 0 0 4 0 0 0 . 5 0 0 . 5 10.2778 0.63923 10.2778 4 0 0 4 0 0 4 0 0 1 2 . 7 7 7 8 4 0 0 4 0 .1 1 1 A2 A1 Var[data]n 9 4 0 0 A2 A1 Var[data]n 4 0 0 4 0 0 4 0 0 400 400 400 0.5 0 4 0 0 4 0 0 18.778 21.778 0 .1 1 1 1 1 3 .4 4 4 9 9 1 6 9 0 1 1 0 4 4 A3 4 0 0 4 0 0 4 0 0 A3 A2 A1 Var[data]n 4 0 0 4 0 0 1 .7 7 7 8 0 .1 1 1 1 13.444 7.1111 40.111 7 .1 1 1 1 Table D6.2: Kalman Filter for testset 54-2200 Table D6.3: Kalman Filter for test set 47-1200 Table D6.4: Kalman Filter for test set 48-1200 2 0 2 0 2 0 23 27 2 0 20 2 0 4 9 4 9 4 9 18.6667 21.6667 Average A3 16.6667 Average 0 2 0 0 2 0 4 0 0 0 0 2 0 4 0 0 1 7 1 6 1 7 25 21 0 art1 art1 0 19 0 0 0 art2 art1 Average 13 27 11 21 14.6667 0 0 0 2322 23 26 0 0 13 1 7 0 0 1 7 1 7 1 7 0 0 20 20 27 26 28 24 20 24 23 21 1 7 1 7 19 20 13 14 23 8 : 3 3 8 : 3 3 8:36 8:59 8:43 8:36 8:398:43 19 23 23 8:39 19 8:59 time trip art3 art2 time trip art3 8:39 8:36 8 : 3 3 8:43 8:59 24 12 time trip art3 art2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 2 33 2 2 22 25 2 2 17 17 Observed Observed 17 14 13 6 16 7.5 10 15 Predict Observed 14.8092 14 0 0 . 5 0.5 9.5 0 . 5 0 . 5 0.5 0 . 5 0 . 5 0.5 a(k+1) Predict a(k+1) 0.05495 16.4945 0.31298 0.39162 0.25743 14.5149 0 0 24.5 e(k+1) e(k+1) 24.5 0.26252 e(k+1) a(k+1) Predict 9.06064 0.37513 1 0.5 0 . 5 0 g(k+D 0 . 50 . 50.5 0 0 g(k+D 0.94505 g(k+D 0 49 0.5 4 0 0 20.5 0.68702 14.084 4 0 0 0 . 5 0 0 . 5 13 0.74257 9.65347 49 4 0 0 25.4444 0.60838 15.48 14.5 0.62487 0.27778 Var[data]n Var[data]n 1 9 9 0.5 4.5 4 9 9 A1 4 0 0 4 0 0 0 . 5 0 1 6 1 .7 7 7 8 1 0 0 9 1 0 0 4 9 4 4 9 A2 A1 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 3 2 . 1 1 1 0.1111 0.1111 1 6 2 5 4 9 2 5 2 5 0 A3 4 0 0 4 0 0 4 0 0 400 400 400 400 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 1 8 .7 7 8 0 . 4 4 4 4 Table D6.5: Kalman Filter for test set 36-1200 Table D6.6: Kalman Filter for test set 16-3000 Table D6.7: Kalman Filter for test set 21-1200 18 16 2 0 2 0 4 0 0 18 2 0 4 9 17 2 0 9 2 0 2 0 15.3333 Average Average A3 A2 A1 Var[data]n 16.3333 Average A3 A2 0 0 1 0 2 0 1 4 1 3 1 4 1 7 0 0 art1 1 7 1 7 1 6 0 0 13 14 17 0 23 21 2 7 1 3 art2 17 16 0 0 0 0 2 0 2 7 0 art3 17 17 1 7 0 19 24 21 20 23 1 7 1 7 0 0 0 2 0 19 13 20 14 trip 1 7 0 0 1 7 19 20 17 trip art3 art2 art1 8 : 3 38:36 8:39 1 7 8:43 8:59 time 8 : 3 3 8:36 8:59 24 11 8:43 8:39 time trip art3 art2 art1 8 : 3 3 8:39 8:43 8:36 8:59 24 17 time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 2 2 2 2 2 14 13 23 12 14 Observed Observed 7 11 17 14 23 23 13.5 14.641 16 Predict Observed Predict Predict 27.2094 33 0.5 9.5 0 .5 0 . 5 a(k+1) 0.4718 a(k+1) 0.17373 0.01274 15.9363 00 0 . 5 0 . 5 8.5 e(k+1) 8.2627 e(k+1) e(k+1) a(k+1) 0.1097 37.5607 1.27684 0.48927 6.05754 0.12771 0 . 5 0.5 0.5 8.5 0.5 g(k+D 0.5282 g(k+D 0.87229 0.52617 40.486 0.47383 0.98726 17 10 0.82627 17 17 0.5 8.5 0.5 4 0 0 2.5 0.51073 4 0 0 0 . 5 0 4 0 0 0 . 5 0 71.1111 76.9444 6.94444 0.11111 Var[data]n Var[data]n Var[data]n g(k+D 9 1 A1 A1 4 0 0 1 8 .7 7 8 0 . 4 4 4 4 4 4 2 5 A2 4 0 0 28.444 28.444 9 4 1 1 6 4004 0 0 400 4 0 0 400 4 0 0 400 0.5 400 400 400 400 0.5 0 0.5 4 0 0 1 1 3 .7 8 113.78 40.111 11.111 2.7778 2.7778 0.1111 0.1111 Table D6.8: Kalman Filter for test set 19-1200 Table D6.9: Kalman Filter for test set 45-1200 Table D6.10: Kalman Filter for test set39-1200 19 2 0 20 400 400 400 400 0.5 0 0.5 15 20 400 400 400 22.3333 16.6667 26.6667 20.3333 Average A3 A2 1 7 1 7 1 7 1 21 6 2 0 2 5 9 6 4 1 4 2 2 2 53 1 2 0 9 2 5 2 5 art1 0 0 2 0 17 17 art2 art2 art1 Average A3 A2 A1 art2 art1 Average A3 0 23 15 17 art3 art3 1 7 0 0 0 2 0 19 23 15 1 7 0 0 0 2 0 1 7 1 7 0 0 0 20 33 19 17 25 19 20 16 33 24 17 22 20 17 17 trip 8:338:36 8:39 178:59 0 24 0 0 20 8:43 time trip art3 8 : 3 3 1 7 0 0 0 8:36 8:39 8:43 8:59 8:33 8:43 8:39 8:59 24 17 14 8:36 time trip time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 2 2 2 2 2 14 12 22 22 21 28 25 20 Observed 9 18 22 21 12 9.5 24.4832 0 . 5 0 . 5 0 .5 0 . 5 0.5 11 0.32 17.96 a(k+1) Predict Observed 0.41064 0.45216 19.2608 0 0 0 . 5 e(k+1) a(k+1) Predict e(k+1) e(k+1) a(k+1) Predict Observed 3.02222 19.0168 0.34299 41.4228 21.2882 0.2969 0 . 5 0 0 .5 0.5 5 0.5 0 . 5 0.5 26.5 0.5 16 0.5 0.68 9(k+1) 9(k+1) g (k+ i) 0.7031 0.54784 0.65701 10 53 32 4 0 0 0 . 5 4 0 0 4 0 0 0 . 5 0 4 0 0 0 . 5 0 6.94444 0.58936 4.09275 4.44444 11.1111 0.73057 8.11742 0.26943 75.6111 30.2778 28.9444 Var[data]n 9 A1 Var[data]n 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 1 0 6 .7 8 6 4 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 400 400 400 0.5 0 1.7778 1.7778 44.444 13.444 4 1 6 1 6 A3 A2 A1 A3 A2 A1 Var[data]n 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 11.111 2.7778 2.7778 7 .1 1 1 1 1 0 6 .7 8 32.111 28.444 0.1111 21.778 0.4444 28.444 44.444 13.444 Table D 6.ll: Kalman Filter for test set 38-1200 Table D6.13: Kalman Filter for test set 29-1200 Table D6.12: Kalman Filter for test set 64-2200 2 0 2 0 2 0 18.3333 18.6667 17.3333 17.6667 20.6667 20.6667 0 0 0 2 0 1 7 1 6 2 0 1 7 1 7 1 7 1 7 2 0 0 2 0 2 0 8 1 2 5 0 2 3 3 1 art1 Average A3 A2 0 0 2 0 0 0 0 0 0 0 2 0 0 17 16 12 17 17 14 art2 art1 Average 0 0 0 2 2 21 11 15 13 31 art3 art2 1 7 0 1 7 19 1 7 0 1 7 1 7 20 19 19 20 12 24 23 20 20 14 8:43 8 : 3 3 8:39 8:59 24 22 17 8:36 time trip 8:36 8 : 3 3 8:39 8:59 24 8 : 3 38:36 1 7 0 8:43 8:59 8:39 8:43 time trip art3 art2 art1 Average time trip art3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 2 2 2 2 2 15 17 17 17 17 20 Observed Observed 12 24 17 24 22 20 Predict Predict 21.2912 26.3353 0 . 5 0.5 15 0 . 5 0 . 5 0.5 11 a(k+1) 0.10689 0.40696 0 0 0 . 5 0 0 . 5 18.25 15.544 0.464 24.288 22 e(k+1) a(k+1) e(k+1) a(k+1) Predict Observed e(k+1) 18.3884 0.43323 2.23277 3.09918 0.14177 0.5 4.5 0 . 5 0 0 . 5 0.5 10 0.5 0 . 5 0.5 g(k+D 9(k+1) 9(k+1) 0.85823 0.59304 4.01951 9 29 0.536 20 2.5 0.89311 4 0 0 0 . 5 0 0 . 5 4 0 0 4 0 0 0 . 5 36.5 36.4444 0.58789 21.4252 0.41211 3.61111 32.4444 0.56677 6.77778 Var[data]n Var[data]n 0 A1 Var[data]n 1 .7 7 7 8 1 3 .4 4 4 1 1 .1 1 1 6 4 4 0 0 4 0 0 2 8 .4 4 4 5 .4 4 4 4 0 .1 1 1 1 9 9 6 4 1 4 1 4 3 6 4 9 4 9 9 1 6 A3 A2 A1 400 400 400 400 0.5 0 4 0 0 400 400 400 400 4 4 . 4 4 4 1 .7 7 7 8 1 3 .4 4 4 53.778 11.111 113.78 Table D6.16: Kalman Filter for test set 8-3000 Table D6.15: Kalman Filter for test set 56-2200 20 400 400 400 24 20 400 400 400 400 2 0 2 0 26 Table D6.14: Kalman Filter for test set 6-3000 23.6667 24.3333 25.6667 25.3333 Average A3 A2 Average A3 A2 A1 0 2 5 2 8 2 8 2 0 3 5 2 5 2 2 2 0 2 2 2 0 2 2 art1 Average art1 art1 0 0 0 0 27 29 23 21 28 26 28 art2 art2 0 0 0 0 20 400 400 400 0 17 17 17 17 25 2 2 27 28 23 29 25 art3 1 7 0 0 0 2 0 1 7 0 0 1 7 19 20 1 7 0 0 0 2 0 19 1 7 19 24 20 24 trip trip art3 art2 trip art3 8:33 8:43 8:59 24 8:36 8:39 8 : 3 3 1 7 8:36 time 8:39 8:43 8:59 time 8:43 20 8 : 3 3 8:36 8:39 8:59 time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 2 13 14 11 26 29 23 2 2 29 Observed Observed 16 16 7.5 22 27 Predict 15.6207 18.0256 0.5 2 0 . 5 2 0.5 0 . 5 0.5 6.5 0 . 5 0 . 5 0.5 9.5 25 a(k+1) a(k+1) Predict 0.34483 a(k+1) Predict Observed 0.12821 0.45087 0 0 0 . 5 2.25 21.25 e(k+1) e(k+1) 13.6046 0.23474 40.1775 0.435 19.2201 e(k+1) 5.49133 2.17949 0.5 0 0 . 5 0.5 0.5 0 0.5 0 . 5 0 0.565 9(k+1) g(k+D 0.65517 1.63793 0.76526 g (k+ i) 2.5 4.5 4 0 0 4 0 0 0 . 5 4 0 0 4 0 0 42.5 10 0.54913 29 0.5 14.5 2.5 0.87179 4 0 0 17.7778 6.94444 0.55275 3.83851 0.44725 71.1111 Var[data]n Var[data]n 8 1 9 4 A1 4 0 0 7 .1 1 1 1 2 8 . 4 4 4 9 4 9 4 4 1 A2 A1 Var[data]n A2 4 0 0 1 9 4 9 4 1 6 1 4 1 400 400 400 400 0.5 0 4 0 0 8 1 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 4 0 0 13.444 0.4444 18.778 7.1111 28.444 28.444 113.78 16 15 2 0 2 0 Table D6.17: Kalman Filter for test set 49-1200 Table D6.18: Kalman Filter for test set 59-2200 Table D6.19: Kalman Filter for test set 24-1200 16 20 400 400 400 20 20 20 Average A3 A2 A1 14.6667 19.3333 21.3333 0 1 7 1 7 1 7 0 0 art1 1 7 1 7 1 3 2 0 0 11 1 2 1 4 art1 Average A3 13 14 11 0 0 20 400 400 400 0 0 art2 14 22 2 2 art2 0 0 17 17 17 0 0 17 11 14 30 20 17 25 art3 1 7 1 7 0 0 0 2 0 20 24 trip art3 1 7 1 7 0 0 19 19 20 20 24 12 20 8 : 3 3 8:43 8:36 8:39 19 8:59 time 8 : 3 38:36 1 7 8 : 3 3 1 7 0 0 8:43 8:59 24 8:39 8:36 8:39 8:43 8:59 time trip art3 art2 art1 Average A3 time trip Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 2 2 2 26 23 26 26 14 25 25 Predict Observed 27.2654 29 0 . 5 a(k+1) Predict Observed 0.40777 25 0.26082 0 0 . 5 0 0 . 5 0 0 . 5 12.5 0.5 12.5 e(k+1) a(k+1) e(k+1) 13.6957 0.3913 23 15.8513 39.3298 0.43365 0 .5 0 . 5 0 . 5 g(k+D g(k+D 0.59223 17.9313 0.56635 0.73918 25 0.5 4 0 0 4 0 0 4 0 0 22.5 0.6087 42.5 0.5 21.25 0.5 30.2778 69.4444 21.4444 9 4 0 0 4 0 0 4 0 0 0 . 5 0 2 1 . 7 7 8 2 1 . 7 7 8 4 0 0 4 0 0 4 0 0 4 0 0 4 0 .1 1 1 9 3 6 A3 A2 A1 Var[data]n 4 0 0 4 0 0 1.7778 58.778 40.111 2 . 7 7 7 8 32.111 106.78 Table D6.20: Kalman Filter for test set 25-1200 Table D6.21: Kalman Filter for test set 20-1200 2 0 4 0 0 2 0 2 0 1 4 9 9 20 2 0 20 400 400 400 2 0 4 8 1 3 6 18.6667 18.3333 24.3333 Average A3 A2 A1 Var[data]n 0 0 2 3 2 3 2 5 2 6 2 9 2 3 art1 art1 Average 0 14 13 11 art2 art2 0 0 0 0 23 20 12 2 2 1 7 1 7 1 7 0 0 19 21 19 8:438:59 20 24 20 11 8 : 3 3 8:36 8:39 8:36 time trip art3 8 : 3 3 1 78:59 0 24 0 0 8:39 8:43 20 30 14 time trip art3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 340 Table D6.22: Prediction Error of the 3 Predictors for Alighting Passenger Prediction Developed KF. Na'ive Dcode Scode Actual d1 d2 d3 Model Model Model 54 2200 23 25 18 30 0.087 0.217 0.304348 47 1200 11 12 20 13 0.091 0.818 0.181818 48 1200 14 14 20 16 0 0.429 0.142857 36 1200 13 12 14 12 0.077 0.077 0.076923 16 3000 17 17 15 16 0 0.118 0.058824 21 1200 22 24 17 35 0.091 0.227 0.590909 19 1200 23 24 23 23 0.043 0 0 45 1200 14 12 17 16 0.143 0.214 0.142857 39 1200 13 12 14 12 0.077 0.077 0.076923 38 1200 12 12 21 10 0 0.75 0.166667 64 2200 22 23 18 21 0.045 0.182 0.045455 29 1200 25 23 22 28 0.08 0.12 0.12 6 3000 17 17 24 12 0 0.412 0.294118 56 2200 22 23 24 20 0.045 0.091 0.090909 8 3000 17 17 20 17 0 0.176 0 49 1200 11 12 16 14 0.091 0.455 0.272727 59 2200 27 24 22 29 0.111 0.185 0.074074 24 1200 23 25 16 25 0.087 0.304 0.086957 25 1200 26 25 25 23 0.038 0.038 0.115385 20 1200 26 25 25 28 0.038 0.038 0.076923 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix D7 1. General This appendix includes the databases for the two bus routes and the prediction results of the developed model and the reference predictors. 2. Database for Route 1 and Route 95 The databases were retrieved from the APC-AVL system embedded on the buses running on the two bus routes. Table D7.1 Database for route 1 (seconds) RCODE 8:59 trip 8:47 trip 8:37 trip 8:27 trip 1 877 992 940 818 2 762 914 900 713 3 929 903 1003 789 4 945 911 1080 922 5 810 800 923 850 6 1029 917 900 880 7 1030 950 946 915 8 914 968 1016 726 9 948 1074 890 817 10 934 901 873 814 11 900 997 956 794 12 986 1013 960 846 13 863 1062 1003 951 14 1122 846 920 855 15 905 958 954 933 16 928 961 932 814 17 1017 844 959 858 18 923 869 1083 809 19 909 1040 919 893 20 840 926 862 842 21 945 975 1001 887 22 793 947 896 895 23 984 879 968 852 341 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 342 24 1070 886 917 912 25 853 980 1006 1000 26 998 850 1008 889 27 1035 1012 950 856 28 768 992 807 913 29 783 862 975 861 30 971 914 1088 789 31 880 827 1043 887 32 982 732 844 857 33 994 1031 959 840 34 984 890 913 936 35 1038 969 887 972 36 980 960 870 825 37 1029 940 987 863 38 805 844 932 807 39 916 850 970 899 40 903 903 937 711 41 767 827 930 918 42 946 840 994 871 43 819 1076 1021 873 44 1050 944 1127 932 45 842 824 861 840 46 967 935 887 849 47 938 833 1045 779 48 831 812 935 754 49 714 914 859 814 50 912 1028 989 874 51 823 950 1085 824 52 975 1000 1020 775 53 965 829 1116 923 54 1076 895 1004 813 55 973 914 1108 856 56 857 825 939 857 57 953 815 878 842 58 823 1026 949 882 59 916 939 1057 841 60 913 878 880 754 61 918 1014 1014 793 62 888 960 919 872 63 1020 846 1008 779 64 742 1064 881 834 65 958 966 972 799 66 1001 1077 881 784 67 986 1045 927 794 68 972 874 1001 832 69 921 817 941 868 70 981 932 1060 850 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 343 Table D7.2: Database for route 95 8:59 8:43 8:40 8:37 B1 B2 A1 A2 A3 632 611 702 618 6 11 11 13 11 639 699 594 529 3 22 10 11 12 571 333 631 553 9 9 9 10 11 653 465 730 622 6 9 8 7 9 487 557 563 720 4 16 14 11 10 667 592 637 567 8 7 11 12 9 608 484 606 766 8 2 10 10 11 521 569 528 547 1 3 11 14 8 542 417 411 687 4 12 10 11 11 619 520 422 499 6 3 10 12 9 620 509 562 541 10 17 9 12 11 554 635 544 579 5 9 12 13 10 584 486 558 646 7 8 10 14 10 684 449 488 804 9 10 9 12 11 583 450 602 635 3 6 12 14 11 610 567 601 585 1 3 17 23 15 651 465 536 674 13 12 19 18 15 656 700 638 510 6 16 15 22 15 616 597 473 615 11 18 17 17 15 521 571 513 668 13 9 15 22 17 519 524 569 596 8 13 18 17 15 631 571 497 609 9 8 17 17 13 568 674 671 642 5 16 15 19 17 639 316 689 665 5 7 17 19 17 533 669 602 786 2 8 16 19 17 633 502 489 816 10 13 13 17 15 642 540 632 692 10 8 15 21 17 629 628 410 494 11 8 17 19 13 553 598 549 660 4 1 14 18 15 589 378 807 570 6 4 17 19 19 632 389 616 524 8 11 13 15 15 522 597 507 692 3 4 17 19 14 570 448 647 671 6 4 18 17 18 550 376 567 606 7 8 17 18 15 662 503 658 588 7 4 9 9 7 701 500 561 540 5 3 10 11 9 646 423 679 448 11 13 8 9 8 568 428 656 468 9 1 9 9 7 620 568 601 512 4 5 8 8 9 596 348 566 585 9 10 9 11 6 562 671 604 642 6 5 8 9 9 627 606 573 647 3 7 8 9 7 614 512 581 506 12 11 7 9 9 547 537 634 692 6 8 9 11 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 344 571 449 531 666 8 25 8 11 8 593 435 531 764 1 2 7 9 9 643 577 586 576 6 14 9 11 9 541 630 536 624 12 12 7 9 9 576 667 483 574 8 7 8 9 7 582 527 646 694 4 3 17 19 15 661 601 568 729 8 7 19 19 15 703 735 478 673 5 7 19 14 17 685 766 559 891 3 6 15 18 15 657 502 576 609 14 4 15 21 13 626 763 425 666 8 19 15 15 13 612 523 492 436 7 4 16 16 13 527 469 461 572 7 8 14 17 15 585 596 663 443 12 9 18 19 17 607 534 576 614 9 13 17 15 15 620 656 663 498 11 11 13 20 15 554 605 625 560 3 9 14 15 18 572 384 619 538 6 3 17 17 15 638 236 539 610 8 14 15 14 15 544 593 606 716 9 18 13 15 14 566 438 494 774 8 3 14 6 5 623 630 536 666 12 4 16 22 19 691 698 540 571 6 6 18 26 21 551 441 600 642 16 8 21 23 17 642 640 462 551 3 1 18 23 18 565 555 490 534 17 3 21 23 23 591 481 600 490 5 7 17 18 19 576 697 688 442 6 2 21 23 18 676 488 611 618 6 12 23 22 23 654 662 556 689 2 7 21 23 18 494 698 583 645 4 8 11 12 8 589 330 592 624 8 18 13 13 11 638 834 721 627 9 2 10 12 10 524 675 581 571 13 18 11 12 9 597 644 651 806 7 14 10 10 11 585 463 581 604 8 6 11 14 8 635 514 531 489 6 9 10 11 11 561 402 608 596 15 17 10 12 9 612 413 659 570 4 9 9 12 11 580 450 471 613 6 10 12 13 10 532 598 511 587 9 10 10 14 10 587 724 573 544 5 6 9 12 11 583 674 618 458 10 16 12 14 11 592 490 522 582 9 14 9 12 11 592 322 541 476 10 19 10 12 9 522 563 629 616 7 4 22 24 18 667 466 494 712 12 11 23 23 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 345 680 361 658 852 4 12 23 18 22 601 584 646 675 9 9 19 23 19 568 597 542 664 14 16 19 26 17 620 625 632 726 8 11 18 18 17 586 513 675 666 7 6 20 20 16 612 498 608 749 1 4 18 22 19 513 519 645 607 10 6 23 24 21 699 522 595 637 8 9 21 19 19 626 561 587 562 2 3 17 25 18 For each trip, APC data of the whole year of observations were processed so that the mean and standard deviation can be used. The following tables provided by the OC Transpo are shown for the sake of illustration. Table D7.3 OC Transpo data for bus trips to Gladstone during 8:30-9:00 AM PROCESSED: 2006-06-20 11:12:12 AUTOMATIC PASSENGER COUNTING SYSTEM REVENUE TIME UTILIZATION WEEKDAY SERVICE BOOKING: JAN05 FROM STOP: RA945 BILLINGS BRIDGE 4C PERIOD: 2005-01-03 TO 2005-04-23 TO STOP: CH080 BANK GLADSTONE NS DAY TYPE: WEEKDAY ROUTE: 1 SOUTH KEYS - OTTAWA ROCKLIFFE TIME: 08:30:00 TO 08:45:00 Average Time per Trip Std % of Minutes Dev Time Moving between stops 7.03 3.17 43.88 Stop and go time 1.46 2.07 9.11 Idle time 1.02 0.98 6.40 Dwell time 2.98 0.85 18.63 Excess time 3.52 0.80 21.99 TOTAL 16.02 1.28 100% Layover time 0. 91 Average sched time per trip 10.00 Total distance (KM) 3.67 Average moving speed (KM/HR) 31.33 Average total speed (KM/HR) 13.75 Total trips captured: 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 346 PROCESSED: 2006-06-20 11:24:24 AUTOMATIC PASSENGER COUNTING SYSTEM REVENUE TIME UTILIZATION WEEKDAY SERVICE BOOKING: SEP05 FROM STOP: RA945 BILLINGS BRIDGE 4C PERIOD: 2005-09-05 TO 2005-12-17 TO STOP: CH080 BANK GLADSTONE NS DAY TYPE: WEEKDAY ROUTE: 1 Bank - - Rockcliffe TIME: 08:30:00 TO 08:45:00 Average Time per Trip Std % of Minutes Dev Time Moving between stops 10.52 1.02 65.78 Stop and go time 0.00 0.00 0.00 Idle time 0.22 0.43 1.36 Dwell time 2. 45 0.79 15.35 Excess time 2.80 0.69 17 .51 1.66 10 0 % Layover time 0.00 Average sched time per trip 10.00 Total distance (KM) 3.67 Average moving speed (KM/HR) 20.94 Average total speed (KM/HR) 13.77 Total trips captured: PROCESSED: 2006-06-20 11:12:32 AUTOMATIC PASSENGER COUNTING SYSTEM REVENUE TIME UTILIZATION WEEKDAY SERVICE BOOKING: JAN05 FROM STOP: RA945 BILLINGS BRIDGE 4C PERIOD: 2005-01-03 TO 2005-04-23 TO STOP: CH080 BANK GLADSTONE NS DAY TYPE: WEEKDAY ROUTE: 1 SOUTH KEYS - OTTAWA ROCKLIFFE TIME: 08:45:00 TO 09:00:00 Average Time per Trip Std % of Minutes Dev Time Moving between stops 10.21 1.85 66.28 Stop and go time 0.29 1.21 1.91 Idle time 0.45 0.60 2.94 Dwell time 2.06 0.54 13.37 Excess time 2.39 0.86 15.49 TOTAL 15.41 1.56 100% Layover time 7.20 Average sched time per trip 10.00 Total distance (KM) 3.67 Average moving speed (KM/HR) 21.56 Average total speed (KM/HR) 14.29 Total trips captured: 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 347 PROCESSED: 2006-06-20 11:15:15 AUTOMATIC PASSENGER COUNTING SYSTEM REVENUE TIME UTILIZATION WEEKDAY SERVICE BOOKING: SEP05 FROM STOP: RA945 BILLINGS BRIDGE 4C PERIOD: 2005-09-05 TO 2005-12-17 TO STOP: CH080 BANK GLADSTONE NS DAY TYPE: WEEKDAY ROUTE: 1 Bank - - Rockcliffe TIME: 08:45:00 TO 09:00:00 Average Time per Trip Std % of Minutes Dev Time Moving between stops 9.08 1.73 59.26 Stop and go time 0.65 1.17 4.22 Idle time 0.83 0.71 5.45 Dwell time 1.85 0.60 12.10 Excess time 2.91 0.87 18.97 TOTAL 15.32 1.54 100% Layover time 2.17 Average sched time per trip 10.00 Total distance (KM) 3.67 Average moving speed (KM/HR) 24.26 Average total speed (KM/HR) 14.38 Total trips captured: 12 Table D7.4 OC Transpo data for bus trips to Mackenzie during 8:00 -10:00 AM PROCESSED: 2 0 0 5 - 0 9 -1 4 1 0 :2 2 :0 1 AUTOMATIC PASSENGER COUNTING SYSTEM POINT CHECK WEEKDAY SERVICE BOOKING: APR05 STOP: CD910 MACKENZIE KING 2A PERIOD: 2005-04-24 TO 2005-06-25 ROUTE: 95 Transitway TIME: 0 8 : 0 0 :0 0 TO 1 0 :0 0 :0 0 DIRECTION: 1 WBND DAY OF THE WEEK: MON TUE WED THU FRI T I M E LOAD AT RUN PAT EXPECTED ACTUAL DIFFERENCE ONS OFFS DEPARTURE DATE 95020A 2 0 8 :0 1 08 03 2 5 10 42 2005-05-10 TUE 08 04 3 11 11 54 2005-05-30 MON AVERAGE 3 8 11 48 (2) 95024A 3 08:04 08 06 2 11 4 47 2005-04-25MON 08 06 2 3 8 50 2005-05-05 THU 08 06 2 4 9 31 2 0 0 5 - 0 6 -1 5 WED AVERAGE 2 6 7 43 (3) 95052A 2 0 8 :0 8 08 11 3 3 9 28 2005-05-19 THU 9S054A 3 08:11 0811 0 5 3 21 2005-05-04 WED 08 12 1 8 10 40 2005-05-30 MON 08 14 3 6 9 61 2005-05-31 TUE 08 12 1 7 11 69 2005-06-07 TUE 08 17 6 8 10 52 2 0 0 5 - 0 6 -1 7 FRI 08 14 3 7 11 42 2005-06-21 TUE AVERAGE 2 7 9 48 (6) 9S010A 2 0 8 :1 4 08 19 5 10 12 82 2 0 0 5 - 0 5 -1 7 TUE 08 18 4 9 11 46 2005-06-06 MON 08 19 5 3 11 53 2005-06-08 WED AVERAGE 5 7 11 60 (3) 95003A 3 0 8 :1 7 08 20 3 5 12 53 2005-04-25 MON 08 21 4 1 9 49 2005-04-27 WED 08 17 0 2 5 39 2005-05-03 TUE 08 19 2 9 7 59 2005-05-12THU 08 21 4 4 9 21 2005-05-17 TUE AVERAGE 3 4 8 44 (5) 95056A 2 08:20 0823 3 10 7 60 2005-05-18 WED Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 348 0 8 : 23 3 17 9 53 2005-06-02 THU AVERAGE 3 14 8 57 (2) 95012A 3 08:24 0 8 : 26 2 17 5 48 2005-05-10 TUE PROCESSED: 2005-09-14 10:22:02 AUTOMATIC PASSENGER COUNTING SYSTEM POINT CHECK WEEKDAY SERVICE BOOKING: APR05 STOP: CD910 MACKENZIE KING 2A PERIOD: 2005-04-24 TO 2005-06-25 ROUTE: 95 T r a n s itw a y TIME 08 :00:00 TO 10:00:00 DIRECTION: 1 WBND DAY OF THE WEEK: MON TUE WED THU F r T I M E LOAD AT RUN PAT EXPECTED ACTUAL DIFFERENCE ONS OFFS DEPARTURE DATE 67 9A 14 08:26 08 30 4 4 5 33 2005-05-18 WED 95055A 2 0 8 :2 7 08 28 1 8 8 35 2005-05-17 TUE 08 28 1 0 3 26 2005-06-03 FRI AVERAGE 1 4 6 31 (2) 95057A 3 08:30 0832 2 6 16 41 2005-05-03 TUE 08 32 2 11 16 58 2005-05-06 FRI 08 36 6 9 14 48 2005-05-10 TUE 08 32 2 7 10 56 2005-06-15 WED AVERAGE 3 8 14 51 (4) 95026A 2 08:33 0836 3 12 8 69 2005-05-18 WED 08 36 3 8 7 51 2005-05-20 FRI 08 37 4 9 8 66 2005-06-16 THU AVERAGE 3 10 8 62 {3) 95005A 3 0 8 :3 6 08 38 2 3 8 26 2005-05-06 FRI 08 38 2 2 9 31 2005-05-17 TUE 08 39 3 5 16 58 2005-05-19 THU AVERAGE 2 3 11 38 (3) 680A 14 08:38 08 44 6 1 1 26 2005-05-09 MON 95009A 3 0 8 :4 0 08 43 3 8 6 69 2005-04-26 TUE 08 41 1 0 9 36 2005-04-27 WED 08 43 3 8 16 66 2005-05-02 MON 08 43 3 3 11 50 2005-05-03 TUE 08 42 2 9 13 62 2005-05-04 WED AVERAGE 2 6 11 57 (5) 95060A 2 08:43 08 45 2 11 4 45 2005-04-26 TUE 08 46 3 4 15 56 2005-05-10 TUE 08 50 7 1 22 45 2005-05-25 WED 08 43 0 6 11 55 2005-06-09 THU 08 44 1 2 10 43 2005-06-20 MON PROCESSED: 2005-09-14 10:22:02 AUTOMATIC PASSENGER COUNTING SYSTEM POINT CHECK WEEKDAY SERVICE BOOKING: APR05 STOP: CD910 MACKENZIE KING 2A PERIOD 2005-04-24 TO 2005-06-25 ROUTE: 95 T r a n s itw a y TIME: 08:00:00 TO 10:00:00 DIRECTION: 1 WBND DAY OF THE WEEK: MON TUE WED THU FRI T I M E LOAD AT RUN PAT EXPECTED ACTUAL DIFFERENCE ONS OFFS DEPARTURE DATE 95060A 2 08:43 AVERAGE 3 5 12 49 (5) 95001A 3 08:46 08 47 1 4 13 22 2005-04-28 THU 659A 14 08:47 0851 4 5 7 30 2005-05-17 TUE 08 46 -1 1 5 22 2005-06-24 FRI AVERAGE 2 3 6 26 (2) 95008A 3 0 8 :4 9 08 : 51 2 4 13 55 2005-04-29 FRI 08 : 52 3 3 8 31 2005-05-24 TUE AVERAGE 3 4 11 43 (2) 9S019A 2 08:53 08 54 1 5 10 41 2005-04-26 TUE 08 53 0 8 12 53 2005-05-03 TUE 08 55 2 7 8 61 2005-05-04 WED 08 50 -3 3 8 29 2005-05-31 TUE AVERAGE 0 6 10 46 (4) 95014A 3 0 8 :5 7 08 : 59 2 6 18 41 2005-05-06 FRI 08 : 57 0 3 10 24 2005-05-09 MON Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 349 0 8 :5 6 -1 8 8 30 2 0 0 5 - 0 5 -1 7 TUE AVERAGE 0 6 12 32 {3} 95053A 4 0 8 :5 9 0 9 :0 4 5 6 14 45 2005-04-27 WED 0 9 :0 6 7 4 12 37 2005-04-29 FRI 0 9 :0 3 4 0 7 21 2 0 0 5 - 0 5 -1 3 FRI 0 8 :5 8 - 1 0 5 45 2005-06-02 THU 0 8 :5 4 -5 1 2 28 2005-06-06 MON AVERAGE 2 2 8 35 (5) 95015A 3 09:01 09:01 0 4 10 51 2005-05-09 MON 0 9 :0 1 0 6 8 36 2005-05-10 TUE AVERAGE 0 5 9 44 (2) 9 5 0 2 3A 2 09:05 09:05 0 5 4 47 2005-05-09 MON 0 9 :0 4 -1 8 8 34 2005-05-10 TUE PROCESSED: 2 0 0 5 - 0 9 -1 4 1 0 :2 2 :0 2 AUTOMATIC PASSENGER COUNTING SYSTEM POINT CHECK WEEKDAY SERVICE BOOKING: APR05 STOP: CD910 MACKENZIE KING 2A PERIOD 2005-04-24 TO 2005-06-25 ROUTE: 95 T r a n s i t w a y TIME 0 8 : 0 0 :0 0 TO 1 0 :0 0 :0 0 DIRECTION: 1 WBND DAY OF THE WEEK: MON TUE WED THU FRI T I M E LOAD AT RUN PAT EXPECTED ACTUAL DIFFERENCE ONS OFFS DEPARTURE DATE 95023A 2 0 9 :0 5 0 9 :0 8 3 23 9 59 2005-05-17 TUE AVERAGE 1 12 7 47 (3) 95050A 3 09:09 09:10 1 2 8 25 2005-05-18 WED 0 9 :1 0 1 5 13 41 2005-05-19 THU 0 9 :1 1 2 7 20 41 2005-05-20 FRI 0 9 :1 1 2 2 6 33 2005-05-26 THU AVERAGE 2 4 12 35 (4) 9 5 0 2 5A 3 09:13 09:16 3 2 15 37 2005-05-25 WED 0 9 :1 5 2 9 6 32 2005-05-26 THU 0 9 :1 3 0 6 7 25 2005-06-02 THU 0 9 :1 8 5 12 17 74 2 0 0 5 - 0 6 -1 3 MON AVERAGE 3 7 11 42 t4) 95018A 10 09:16 09:21 5 10 20 47 2005-04-25 MON 0 9 :1 8 2 11 7 32 2005-04-26 TUE 0 9 :1 9 3 1 11 47 2005-04-27 WED 0 9 :1 6 0 1 10 25 2005-04-28 THU 0 9 :1 6 0 3 7 19 2005-05-05 THU AVERAGE 2 5 11 34 (5) 95021A 2 0 9 :2 0 0 9 :2 2 2 10 15 40 2005-05-30 MON 95004A 3 0 9 :2 4 0 9 :2 6 2 5 17 35 2005-04-26 TUE 0 9 :2 6 2 6 3 20 2005-04-27 WED 0 9 :2 3 -1 6 2 24 2005-04-28 THU 0 9 :2 8 4 6 18 51 2005-05-05 THU AVERAGE 2 6 10 33 (4) 95022A 3 0 9 :2 8 0 9 :3 1 3 0 5 16 2005-05-11 WED 0 9 :3 1 3 5 9 26 2005-05-12 THU AVERAGE 3 3 7 21 (2) 95007A 2 09:32 09:33 1 9 13 44 2005-05-20 FRI PROCESSED: 2 0 0 5 - 0 9 -1 4 1 0 :2 2 :0 3 AUTOMATIC PASSENGER COUNTING SYSTEM POINT CHECK WEEKDAY SERVICE BOOKING: APR05 STOP: CD910 MACKENZIE KING 2A PERIOD : 2005-04-24 TO 2005-06-25 ROUTE: 95 Transitway TIME 08:00:00 TO 10:00:00 DIRECTION: 1 WBND DAY OF THE WEEK: MON TUE WED THU FRI T I M E LOAD AT RUN PAT EXPECTED ACTUAL DIFFERENCE ONS OFFS DEPARTURE DATE 95028A 3 09:36 09:39 3 6 12 36 2005-05-26 THU 95013A 3 0 9 :4 0 0 9 :4 4 4 5 15 21 2005-05-03 TUE 0 9 :4 3 3 1 4 15 2005-05-04 WED AVERAGE 4 3 10 18 (2) 95011A 9 09:44 09:45 1 8 5 19 2005-04-25 MON 0 9 :4 9 5 9 17 40 2005-04-27 WED AVERAGE 3 9 11 30 (2) 95017A 3 0 9 :4 8 0 9 :4 8 0 6 11 37 2 0 0 5 - 0 5 -0 6 FRI 0 9 :4 7 -1 5 15 18 2005-05-09 MON 09:50 2 10 16 34 2005-05-10 TUE 0 9 :4 7 -1 6 9 29 2005-06-24 FRI AVERAGE 0 7 13 30 (4) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 350 95027A 3 09:52 0 9: 52 0 5 13 34 2005-05-09 MON 0 9: 53 1 6 10 30 2005-06-16 THU AVERAGE 1 6 12 32 (2) 95024A 2 09:56 09: 59 3 1 6 12 2005-04-25 MON 09* 58 2 10 12 41 2005-05-05 THU 09 58 2 9 8 23 2005-06-15 WED AVERAGE 2 7 9 25 (3) 95016A 3 10:00 09 59 -1 5 8 15 2005-05-04 WED 10 02 2 10 6 19 2005-05-09 MON 10 00 0 7 10 24 2005-05-17 TUE 10 02 2 3 26 18 2005-06-02 THU 09 59 -1 3 10 23 2005-06-15 WED AVERAGE 0 6 12 20 (5) PROCESSED: 2 0 0 5 -0 9 -1 4 1 0 :2 5 :4 2 AUTOMATIC PASSENGER COUNTING SYSTEM POINT CHECK WEEKDAY SERVICE BOOKING: SEP04 STOP: CD910 MACKENZIE KING 2A PERIOD: 2004-09-05 TO 2004-12-18 ROUTE: 95 ORLEANS - NEPEAN SOUTH TIME: 08:00:00 TO 10:00:00 DIRECTION: 1 WBND DAY OF THE WEEK: MON TUE WED THU FRI T I M E LOAD AT RUN PAT EXPECTED ACTUAL DIFFERENCE ONS OFFS DEPARTURE DATE 95015A 3 08:00 08 05 5 4 11 55 2 0 0 4 - 0 9 -2 3 THU 07 59 -1 4 3 15 2004-11-12 FRI AVERAGE 2 4 7 35 (2) 95005A 2 08:03 08 04 1 5 6 28 2004-10-08 FRI 95Q64A 3 08:06 08 13 7 9 9 64 2 0 0 4 - 1 0 -1 8 MON 08 13 7 10 7 71 2004-10-19 TUE 08 07 1 8 4 38 2004-10-26 TUE 08 09 3 8 8 56 2004-10-27 WED 08 14 8 8 9 41 2004-11-01 MON AVERAGE 5 9 7 54 (5) 95063A 2 0 8 :0 9 08 13 4 14 9 54 2004-10-18 MON 95062A 3 08:12 08 19 7 11 14 63 2004-09-17 FRI 08 18 6 7 4 46 2004-10-21 THU AVERAGE 7 9 9 55 (2) 95007A 2 08:15 08 22 7 16 7 60 2004-09-29 WED 0820 5 5 6 39 2004-10-29 FRI 08 20 5 14 10 77 2004-11-09 TUE AVERAGE 6 12 8 59 {3) 95009A 3 08:19 08 22 3 1 2 33 2004-09-10 FRI 08 22 3 8 7 42 2004-11-15 MON AVERAGE 3 5 S 38 (2) 95020A 2 0 8 :2 2 08 28 6 6 12 58 2004-09-07 TUE 08 25 3 9 10 58 2004-11-19 FRI AVERAGE 5 8 11 58 (2) 95019A 3 08:25 N/A 437A 4 0 8 :2 7 08 33 6 0 8 40 2004-10-27 WED PROCESSED: 2 0 0 5 - 0 9 -1 4 1 0 :2 5 :4 2 AUTOMATIC PASSENGER COUNTING SYSTEM POINT CHECK WEEKDAY SERVICE BOOKING: SEP04 STOP: CD910 MACKENZIE KING 2A PERIOD: 2004-09-05 TO 2004-12-18 ROUTE: 95 ORLEANS - NEPEAN SOUTH TIME: 08:00:00 TO 10:00:00 DIRECTION: 1 WBND DAY OF THE WEEK: MON TUE WED THU FRI T I M E LOAD AT RUN PAT EXPECTED ACTUAL DIFFERENCE ONS OFFS DEPARTURE DATE 95056A 2 08:29 08 : 32 3 7 4 43 2004-10-13 WED 08 :28 -1 6 11 39 2004-11-29 MON AVERAGE 1 7 8 41 (2) 95025A 3 08:32 08 : 34 2 7 11 47 2004-11-03 WED 08 : 40 8 5 10 54 2004-11-05 FRI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 351 08 36 4 3 9 36 2004-11-08 MON AVERAGE 5 5 10 46 (3) 95008A 2 0 8 :3 5 08 42 7 7 15 28 2004-11-05 FRI 634A 4 0 8 :3 7 08 38 1 0 5 21 2004-10-01 FRI 95026A 3 0 8 :3 9 08 41 2 4 3 42 2004-09-08 WED 08 42 3 1 7 27 2004-09-10 FRI 08 42 3 5 6 28 2004-09-23 THU 08 40 1 3 6 36 2004-09-24 FRI 08 41 2 2 9 34 2004-12-01 WED 08 41 2 8 6 40 2004-12-06 MON AVERAGE 2 4 6 35 (6) 95018A 3 0 8 :4 2 N/A 95022A 2 0 8 :4 5 08 46 1 8 9 50 2004-09-17 FRI 08 46 1 11 6 48 2004-09-21 TUE 08 48 3 4 15 57 2004-12-06 MON AVERAGE 2 8 10 52 (3) 699A 4 08:47 0847 0 0 0 15 2004-09-21 TUE 08 50 3 1 4 34 2004-09-22 WED 08 53 6 0 7 23 2004-09-24 FRI AVERAGE 3 0 4 24 {3) 95002A 3 0 8 :4 9 08 50 1 3 11 34 2004-09-14 TUE PROCESSED: 2 0 0 5 -0 9 -1 4 1 0 :2 5 :4 3 AUTOMATIC PASSENGER COUNTING SYSTEM POINT CHECK WEEKDAY SERVICE BOOKING: SEP04 STOP: CD910 MACKENZIE KING 2A PERIOD 2004-09-05 TO 2004-12-18 ROUTE: 95 ORLEANS ~ NEPEAN SOUTH TIME 0 8 :0 0 :0 0 TO 1 0 :0 0 :0 0 DIRECTION: 1 WBND DAY OF THE WEEK: MON TUE WED THU FRI T I M E LOAD AT RUN PAT EXPECTED ACTUAL DIFFERENCE ONS OFFS DEPARTURE DATE 95014A 3 0 8 :5 3 N/A 95016A 2 0 8 :5 7 08 59 2 3 5 41 2004-09-10 FRI 09 00 3 9 7 54 2004-09-15 WED 09 : 00 3 11 10 60 2004-09-28 TUE AVERAGE 3 8 7 52 (3) 42 3A 4 0 8 :5 9 09 :0 2 3 1 3 10 2004-11-05 FRI 95052A 3 09:01 09 : 01 0 3 4 32 2004-10-15 FRI 09 : 01 0 1 8 40 2004-10-19 TUE 09 : 01 0 12 11 43 2004-11-10 WED 09 : 01 0 4 7 29 2004-11-16 TUE AVERAGE 0 5 8 36 (4) 95027A 3 0 9 :0 5 09 : 06 1 7 6 41 2004-09-09 THU 09 : 03 -2 2 3 40 2004-10-28 THU AVERAGE -1 5 5 41 (2) 9 5 0 1 3A 2 0 9 :0 8 N/A 95023A 3 0 9 :1 3 N/A 95050A 10 0 9 :1 6 09 : 18 2 4 4 36 2004-10-05 TUE 09 : 17 1 2 8 25 2004-10-28 THU 09 : 17 1 4 8 23 2004-10-29 FRI 09 :24 8 5 18 51 2004-11-05 FRI 09 : 18 2 3 2 32 2004-11-10 WED AVERAGE 3 4 8 33 {5} 95004A 2 09:18 N/A 95055A 2 09:20 09 :2 0 0 4 9 23 2004-11-09 TUE 95010A 3 0 9 :2 4 09 : 28 4 13 9 47 2004-09-21 TUE 09 :2 6 2 2 12 38 2004-09-30 THU Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 352 PROCESSED: 2005-09-14 10:25:43 AUTOMATIC PASSENGER COUNTING SYSTEM POINT CHECK WEEKDAY SERVICE BOOKING: SEP04 STOP: CD910 MACKENZIE KING 2A PERIOD: 2004-09-05 TO 2004-12-18 ROUTE: 95 ORLEANS - NEPEAN SOUTH TIME: 08:00:00 TO 10:00:00 DIRECTION: 1 WBND DAY OF THE WEEK: MON TUE WED THU FRI T I M E LOAD AT RUN PAT EXPECTED ACTUAL DIFFERENCE ONS OFFS DEPARTURE DATE 95010A 3 0 9 :2 4 09:25 1 11 5 32 2004-10-20 WED AVERAGE 2 9 9 39 (3) 95024A 3 0 9 :2 8 N/A 95001A 2 0 9 :3 2 N/A 435A 3 0 9 :3 4 0 9 :3 4 0 1 5 15 2004-09-17 FRI 0 9 :3 6 2 3 5 12 2004-09-21 TUE 0 9 :3 7 3 2 7 20 2004-10-13 WED 0 9 :3 7 3 1 10 18 2004-10-14 THU 0 9 :3 9 5 4 9 36 2004-10-26 TUE AVERAGE 3 2 7 20 (5) 95057A 3 09:36 09:40 4 10 6 49 2004-10-29 FRI 0 9 :3 8 2 5 15 43 2004-11-04 THU 0 9 :3 9 3 3 17 24 2004-11-18 THU AVERAGE 3 6 13 39 (3) 9 5 0 0 6A 3 0 9 :4 0 0 9 :4 3 3 5 12 33 2004-09-14 TUE 95017A 9 0 9 :4 4 N/A 95058A 3 09:46 09:48 2 1 10 19 2004-10-01 FRI 0 9 :4 4 -2 3 17 27 2004-10-07 THU 0 9 :4 9 3 9 10 44 2004-11-01 MON 0 9 :4 9 3 10 8 35 2004-11-03 WED 0 9 :4 9 3 11 14 23 2004-11-08 MON AVERAGE 2 7 12 30 (5) 95012A 3 09:48 0 9 :5 0 2 1 9 19 2004-10-28 THU 0 9 :4 9 1 8 21 25 2004-10-29 FRI AVERAGE 2 5 15 22 (2) 95051A 3 09:50 09:56 6 8 21 30 2004-10-12 TUE 0 9 :5 5 5 4 7 27 2004-11-05 FRI PROCESSED: 2005-09-14 10:25:43 AUTOMATIC PASSENGER COUNTING SYSTEM POINT CHECK WEEKDAY SERVICE BOOKING : SEP04 STOP: CD910 MACKENZIE KING 2A PERIOD: 2004-09-05 TO 2004-12-18 ROUTE: 95 ORLEANS - NEPEAN SOUTH TIME : 08:00:00 TO 10:00:00 DIRECTION: 1 WBND DAY OF THE WEEK: MON TUE WED THU FRI T I M E LOAD AT RUN PAT EXPECTED ACTUAL DIFFERENCE ONS OFFS DEPARTURE DATE 95051A 3 0 9 :5 0 AVERAGE 6 6 14 29 (2) 95021A 3 0 9 :5 2 N/A 95015A 2 09:56 09:56 0 2 8 21 2004-09-23 THU 0 9 :5 6 0 8 8 13 2004-11-12 FRI AVERAGE 0 5 8 17 (2) 95011A 3 1 0 :0 0 N/A Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 353 3. Running Time Prediction Results of the Developed Model :Route 1 The running time predictions obtained from the developed model are tabulated in Table D7.19 of this appendix. For illustration, the following table depicts the output of the model applied for test case RCODE=18. Table D7.5: An Example of Developed Model’s Output the Edist to the first second... neighbor Columns 1 through 12 138.2730 210.8571 87.9496 77.7618 189.6905 163.5890 132.6671 136.3413 224.3460 193.2873 134.9488 129.1932 Columns 13 through 24 175.2300 164.9713 137.6980 136.4368 133.9282 183.4395 198.7468 82.0132 171.2472 106.2629 162.6202 166.4866 Columns 25 through 36 105.1114 136.7360 267.5160 110.4786 69.3883 112.0025 294.2738 141.9854 175.7238 215.7770 193.6065 74.5923 Columns 37 through 48 160.6042 131.7126 187.8599 180.5488 115.3895 151.7052 106.8601 226.4662 173.5288 122.7170 197.4734 205.1065 Columns 49 through 60 121.8729 40.4245 109.3997 137.6448 76.3830 51.7569 161.5609 216.7234 149.0179 12.5770 211.3152 109.9486 Columns 61 through 68 145.7678 122.9194 223.5827 107.5712 239.6461 183.4306 84.3911 166.3349 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 354 bandwidth = 180.5488 The matrix of Error is 1.0e+004 * Columns 1 through 12 0.2593 2.8210 0.0043 0.0005 1.4051 1.0797 0.9566 0.0297 0.0692 0.0196 0.0739 0.3250 Columns 13 through 24 0.5662 3.9312 0.0856 0.0017 0.8072 0.0462 0.6723 0.0004 1.8555 0.2964 2.1052 0.7821 Columns 25 through 36 0.3783 1.1622 2.4488 2.3185 0.0933 0.3824 0.3668 0.4822 0.3481 1.4091 0.3011 0.7268 Columns 37 through 48 1.5127 0.0346 0.0641 2.8624 0.0193 1.4681 1.2478 0.6955 0.2016 0.0056 0.8899 4.3129 Columns 49 through 60 0.0596 1.5289 0.1847 0.0511 1.8846 0.0905 0.5140 0.0841 1.1825 0.0911 0.0183 0.0172 Columns 61 through 69 0.2168 0.9249 3.4738 0.0822 0.6648 0.4521 0.1124 0.0061 0.1337 Cv = 7.7825e+003 The CROSS VALIDATIOn factors 1.0e+003 * 9.6041 8.1329 7.6980 7.9301 7.8487 7.7976 7.7825 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 355 OpNei = 21 B eta- 1.0e+003 * 1.3238 - 0.0000 - 0.0002 - 0.0002 THE PREDCTED BUS RUNNING TIME IS 948.96 SECONDS.Program complete ans = 948.9596 Note: Only the most important part of the output is printed out. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 356 4. Running Time Prediction Results of the KF Predictor: Route 1 The predictions of Kalman Filter predictors are tabulated in Tables D7.6-D7.20. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced m Table D7.6. Kalman Filter for test case RCODE=l ■o o -G CM « CO < f O 5 JC 7 CO > 2 ** r t + + <8 c «# a £ ® ® A as I I time trip A art1 Average A A Predict N» o o s O o B.f7 o 630 396900 396900 396900 396900 0.5 900 | CO 00 CO CM CO ' I 8:17 o 630 396900 396900 396900 396900 0.5 0.5 900 I O S CO 0 0 'O’ to co OCD CO CO o CO CM I I 8:27 817.2 813.5 I 794.1 - h 79.8044 27.3878 200.694 53.5961 0.5 26.7981 847.05 CO CO d CD O CO f T CO CD o ID M- 8:37 CO CO CD CO 872.9 CD CO CO 906.167 274.454 1106.67 CM CO CO CO CO 0.50952 351.853 901.949 960.2 I S006 CM O CO d 0 0 CO in CO 8:47 20 1073.5 996.7 990.233 6933.34 8052.07 41.8178 7492.7 0.51147 3832.3 978.869 CM CO r ^ to I 8:57 24 948.2 933.9 900.1 927.4 432.64 42.25 745.29 0.94487 224.355 0.05513 906.335 1013.2 | 900 844 852.2 984.4 Observed Observed Observed 857 931 931.663 959 888.634 1017 960.507 923 833.079 874.654 967.9 0.5 900 0.5 878.75 808.6 0.5 0.5 897.65 0.48129 959.83 0.47761 958.1970.38312 868.9 0.48501 0.15827 949.971 878.9 0 0 0.5 900 0 1119.7 0.5 773.72 309.46 1612.56 138.847 0.07191 948.278 1083.1 e(k+1) a(k+1) Predict 502.553 241.716 0.17126 e(k+1) a(k+1) Predict e(k+1) a(k+1) Predict 0.5 1781.28 0.5 0.5301 6673.23 0.4699 0.61688 782.259 g(k+i) g(k+i) g(k+i) 0.51499 2578.18 0.84173 2239.4 0.5 396900 0.5 396900 0.5 396900 1268.09 12588.7 149.605 0.92809 3562.57 3890.34 0.53034 2063.19 0.46966 5006.28 Var[data]n 432.64 618.92 0.5 396900 1826.14 3108.78 0.51871 10.6711 291.668 0.82874 5918.74 1481.12 0.52239 A1 Var[data]n 564.854 6664 660.49 6.76 597.05 396900 347.201 7401.73 4431.12 4320.87 2826.69 A2 A1 Var[data]n 30.25 268.96 118.81 1388.8 1573.44 533.61 396900 396900 396900 396900 396900 396900 396900 5309.55 908.018 4194.72 2930.42 113.068 378.951 236.134 2055.11 481.071 4524.8 3348.55 630 630 630 948.2 935.067 917.867 950.133 920.933 868.133 867.167 157.921 Average A3 A2 Table D7.9: D7.9: Table RCODE=17 Kalman for Filter test case Table D7.ll.Table RCODE=23 for Kalman Filter test case Table D7.10: Table D7.10: RCODE=18 Kalman Filter test case for 0 0 814 844 792.5 859.067 946.6 949.2 959.1 1017.4 art1 0 814 857.5 932.9 974.9 0 0 0 0 0 0 919.7 953.7 931.8 953.7958.2 931.8 960.6 841.5 886.7 895.3 874.5 1089 148.84 932.9 839.6 945.1 861.7 1001.2 895.8 919.567 17 17 17 18 20 24 trip art3 art2 art1 Average A3 A2 A1 trip art3 art2 8:17 8:17 8:47 20 845 948 960.6 8:57 24 904.8 928.2 8:17 8:47 8:57 24 1121.9 904.8 928.2 984.967 18750.7 6426.698:27 3222.45 18 8:278:37 18 19 854.6 8:37 19 8:27 8:57 8:47 20 926.1 8:37 19 time trip art3 art2 art1 Average A3 time time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced co O n Table D7.12. Kalman Filter for test case RCODE=28 < 2, o ' 'fO 7 O C < CM < T~ T“ > 'w O P CM t r t: *-* + + ra ID ID c <0 ID 1 1 time trip <0 Average Predict Observed | o © 8:17 s. o o 630 396900 396900 396900 396900 0.5 0.5 900 1 I I 8:27 00 999.7 889.4 856,3 915.133 7151.52 662.204 3461.36 3906.86 0.5 1953.43 I 0.5 878.15 913.4 I 00 o h-’ o CO CO CM 00 I 8:37 ) O 1006.3 1007.7 949.5 987.833 341.018 394.684 1469.44 367.851 317.532 0.13679 944.562 o 00 CO 0 0 N 20 979.9 849.5 1011.5 946.967 9499.75 4164.55 5292.18 0.51456 2723.16 0.48544 912.422 991.6 I 00 lO CM 8:57 24 CO 997.9 1034.9 961.867 11895.5 1298.4 5333.87 6596.97 0.58554 3862.8 0.41446 1016.95 768.4 | <2 b • Pi U O Q W CO fO 2 l"- Q H & « 60 & 4> S Sx a> 08 S3 p II e3 M £ < 5 CM CO < CM CO r + O) I I time trip (0 IB art1 Average Var[data]n e(k+1) a(k+1) Predict Observed o I I 8:17 v h o o o 63 0 396900 396900 396900 396900 0.5 0.5 900 I CO o’ SO CO CO CO Tj- CO o 0 0 o CM lO 0 0 © 5 CM 8:27 CO 788.6 887 856.7 158.76 0.5 1230.17 878.35 839.9 | 0 0 CO CM o t O 0 0 ■M- o> 8:37 1087.5 CO 'M- CO CO 991.567 9203.2 2686.69 21835 5944.95 0.54688 0.45312 842.033 959.1 I 0 0 CM 0 0 I I 8:47 20 913.5 CM 732.2 824.3 7956.64 8482.41 3982.52 0.64493 2568.46 0.35507 812.765 1030.7 | CM 0 0 0 0 8:57 971 d ~ r 982.1 944.4 707.56 4134.49 1421.29 2421.03 0.6733 1630.07 0.3267 997.978 993.9 | a u O © T r- H >D Q - t n © W mj ■w « U V ce O 93 se s 93 93 s O *o n "0 "lO *3 £ < < O C <1 M C C CO t CM 0 + + m • I time trip ID Table D7.15. Kalman Filter for test case RCODE = 40 < < 7 O .Q ■o M < CM CO 7 ■jg + re to © w © time | trip art 3 art1 | Average Var[data]n © re Predict > o o O SO N. o I I 8:17 630 396900 396900 396900 396900 0.5 900 I CO I 8:27 873 932 840 881.667 75.1111 2533.44 1736.11 1304.28 0.5 652.139 0.5 870 848.8 I CM 00 CO o o> O ) O h-i CO o> 8:37 1021.1 1003.13 322.801 15442.2 20230.3 7882.5 0.51986 0.48014 855.09 886.7 I CO ,^r CM 8:47 20 1076.2 943.6 948.067 16418.2 19.9511 15293.4 8219.05 0.59977 4929.56 0.40023 849.334 934.5 I CO 00 CM CM CO CM 00 CM O CM 00 r*- 8:57 24 1050 842.2 903.633 7213.67 3774.05 14318.4 0.57343 0.42657 881.572 967 | H 3 Q h V© * a oc U O Q w TT VO +■> 05 v S3 U V 9 0 O 4> s <3 »> '55 7 *© '« g J > O Q . O " g J 3 2 e 'r £ O C M C £ < g j t t: + + re re 10 9 £ © re I I time trip re re Average A3 Predict o N. o o o I 8:17 630 396900 396900 396900 396900 0.5 0.5 900 I SO SO CO 8:27 00 778.5 754.2 813.5 782.067 12.7211 776.551 988.054 394.636 197.318 856.75 T“ 5 0 O d 00 CM I 8:37 1044.9 935.3 859.1 946.433 9695.68 123.951 7627.11 O 0.50985 2503.27 0.49015 866.599 989.4 | CO CO CO CO 00 o CO o CM CM CO 00 I I 8:47 20 832.8 812.1 913.9 852.933 405.351 1667.36 3716.93 o 0.77352 to 930.999 1027.6 | CM 00 CM 00 d 00 8:57 938.1 831.3 714.4 827.933 12136.7 11.3344 6074.01 0.53095 3225.01 0.46905 861.306 912 | a U H 3 Q ' t ►4 a a: O Q W I/i o ■*o ■M O 85 s 58 0 a> 9 0 4> 58 9 0 O '55 .a "O '© re 'r g J g j < CM < g j CM + + + © © © I 1 1 time trip art3 re art1 1 Average A3 Var[data]n Predict O o o o 8:17 ■s— N. 630 396900 396900 396900 396900 0.5 0.5 900 | CM T— 5 0 ] SO o o T—CO CO o c I 8:27 CO 856 857 841 225 518.5 259.25 0.5 878.5 841.6 I CO O CM CO o> o 00 o CO o> ) O 00 o> CM I 8:37 o 1108.4 939.1 1017.13 175.121 8329.6 4252.36 0.51479 0.48521 891.792 877.7 I CO d CM 00 o o T}- p CO 8:47 CM 895.3 913.8 824.8 877.967 2826.69 792.236 0.79006 625.91 0.20994 835.906 814.9 | 00 CO CO O CM o> CM o CM o> 00 to CO I 8:57 24 1076 972.9 968.733 11506.1 17.3611 12417.4 5761.75 0.52576 0.47424 953.1 | Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced m VO Table D7.18. Kalman Filter for test case RCODE = 57 TS 'T V M C < n < < "3 7 JC T* t + + (0 (0 I time trip art3 Average Var[data]n Predict Observed | o o o to o o tr> T— hs~ o 1 1 8:17 630 396900 396900 396900 396900 900 I CM CM o to O m CO CO CM to QO CM 1^ to I 8:27 881 840 754 | 3136 5041 1680.5 | 840.25 793.4 | 5 0 8:37 949 1056.6 879.5 961.7 161.29 9006.01 6756.84 4583.65 0.54198 2484.25 0.45802 840.065 1014 I o CO CO 00 CO O CM CM Tf- CO GO CM I 8:47 1025.5 939.4 878 947.633 67.7878 3065.5 0.64418 1974.73 0.35582 926.392 1013 I o' CO CO O ■M* CM CO CM CO CO GO 00 Tj- I 8:57 915 913 883.667 981.778 2331.11 0.64877 1512.35 0.35123 948.123 j 917.6 I a u H a o Q W O V Q a E "3 a GA O o § u o ft GA ft s o II n < 'S T* 'o lo £ . < < CO CM T“ r t + + + I time trip <0 CO <0 Average Var[data]n Predict Observed | o 006 o o o to o lO N» I I 8:17 o 630 396900 396900 396900 396900 00 o in o IO CO CM CM 00 I 8:27 798.8 784.4 793.9 792.367 41.3878 63.4678 2.35111 52.4278 26.2139 846.95 5 0 5 0 CM CO CM O CM 00 I 8:37 971.8 881.2 926.767 2076.32 0.28444 2052.16 0.50317 1032.59 0.49683 880.052 1000.7 | CM o CM lO O 5 0 ’M’ CO I I 8:47 965.5 1076.8 1044,9 1029.07 4040.72 2278.47 3159.6 0.57023 1801.69 0.42977 1025.9 874.3 00 CM to 00 O CM o o | 8:57 957.6 986 981.667 579.204 389.404 18.7778 484.304 399.638 0.17482 966.473 971.6 | a W \ r W © v Pi 00 Q t"-’ N H * Q "3 3 CA o 5 U o I! t f GO o 4> CS s v e N < T“ < < O C M C CO tr t r + + + oi CO CO I I time trip CO CO Average Var[data]n Predict j Observed | o IO o IO o o o N. o I 8:17 630 396900 396900 396900 396900 900 | o to o IO 00 I I 8:27 798.8 784.4 793.9 792.367 41.3878 63.4678 2.35111 52.4278 26.2139 846.95 \ 832.2 | 00 00 o CM CM O CM © CM Tf "S’ o in T” 05 00 CO I I 8:37 971.8 881.2 927.3 926.767 2076.32 2052.16 0.50317 1032.59 0.49683 1000.7 | MCM CM o I 8:47 965.5 1076.8 1044.9 1029.07 4040.72 2278.47 250.694 3159.6 0.57023 1801.69 0.42977 1025.9 874.3 | oo in 957.6 1001.4 986 981.667 579.204 389.404 18.7778 484.304 0.82518 399.638 0.17482 966.473 r 971.6 | d3 0.0140 0.0364 0.1294 0.2420 0.3894 0.1517 0.1262 0.1267 0.1441 0 0 2446 0.0778 0.2984 0.1618 0.0651 0.0716 0.0054 0.0338 d1 d2 0.0657 0.0363 0.0502 0.0827 0.0084 0.0776 0.0563 0.1289 0.0153 Error 12.32 0.0729 0.0953 70.50 0.0633 0.0858 69.21 0.0762 Naive -70.56 108.72 0.0035106.10 0.0558 146.58 0.0186 201.87 0.0909 0.1392 0.2272 -240.72 0.0786 0.1130 -124.32 0.0280 0.0394 -246.14 0.1148-388.58 0.2902 0.0828 0.0045 -136.61 KF -6.92 Error 79.17 34.90 -36.25 85.43 50.69 -37.40 128.77 151.32 248.58 -106.90 -248.55 Error 78.41 61.88 70.53 74.65 67.69 62.43 -68.90 -25.86 -88.19 Devlp. 915 NaTve Model 954.68 803.28 3.23 1066.05 1047.42 1014.54 KF 984 1117.82 960.5 Model 888.63 815.53 92.48 997.97 1382.48 82.34 -4.07 881.57 909.82 973.46 837.19 847 53.70 115.91 966.47 825.02 18.07 5.13 833.08 735.82 49.37 967.12 959.79 948.12 1054.21 14.07 -30.52 1016.95 Table D7.21: Prediction the Error Predictors ofD7.21: Table 946 899.4 Model 962.11 994.9 923.07 906.33 924.92 896.47 Developed 967 912 908.77 861.31 1029 950.59 985.5 768.4 858.62 923.1 948.96 984.4 935.03 917.6 903.53 1017.4 1 877.1 7 1029.8 12 17 18 33 993.9 911.56 50 57 953.1 23 28 37 46 40 902.9 828.25 61 68 971.6 953.53 Rcode Actual Note: dl,d2,d3: The absolute relative error (%) ofthe Proposed Model, the Kalman Filter and the Naivepredictor Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 363 5. Running Time Prediction Results of the Developed Model: Route 95 The running time prediction results of the developed model are tabulated in Table D736 of this appendix. 6. Running Time Prediction Results of the KF Model: Route 95 Tables from D7.22-D7.35 present the running time prediction results of KF model applied to route 95. 7. Boarding Passenger Prediction Results of the Developed Model: Route 95 The boarding passenger prediction results of the developed model are tabulated in Table D7.51 in this appendix. 8. Boarding Passenger Prediction Results of the KF Model: Route 95 Tables from D7.37-D7.50 show the predictions results of the KF model applied for boarding passenger predictions. 9. Alighting Passenger Prediction Results of the Developed Model: Route 95 The alighting passenger prediction results of the developed model are tabulated in Table D7.65 of this appendix. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 364 10. Alighting Passenger Prediction Results of the KF Model: Route 95 Tables from D7.52 -D7.64 show the prediction results of the KF model applied for alighting passenger predictions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced cn VO Table D7.22. Kalman Filter for test case RCODE —03 <— o .Q < < '© 'a .t oto to to Q. *D > *C ■o 'a '•S c * CO M < <3 CO 2 4-1 ! t + + • o 0 © 9 t 9 2 a a w 0 c 9 E 9 Ot OCO CO to CO CD © > 8 , 1 0 .5 6 9 ' CM o CD CO CO 00 CO T“ CO CO <0 T“ rv T— 03 to d CM CO rv o Ot to to CO CO to O) t s CO CO o CO CO Sft CO CO to 03 00 CO CO t s CM cd o CO d CO CO d CO o 5 0 st o O I rv o T“ CO CO 00 rv to h- u 1 t f Q w n r * c f w w PS j r • H Q "H © Table D7.25. Kalman Filter for test case RCODE = 20 O *o 're £ .Q CM < co < < T* '© tr + + + © © (Q [ [ time trip ~ art3 H art1 Average Var[data]n Ol Predict u> d to o I I 8:33 N. o o 630 396900 396900 396900 396900 d to o 630 I d in I 8:36 00 co b- CO 00 509.6 | 615.47 599.623 5502.18 8104.2 251.117 CD 00 o CO 3 0 d in 3401.59 622.735 667.52 I CO IO CM IO CM o CO CO 8:39 3 0 536.49 638.44 472.78 549.237 162.478 7957.23 5845.62 4059.86 0.64762 2629.25 O 513.23 I MCM CM o CM CO CO V CO 00 CM in 00 I 8:43 464.78 699.71 596.78 587.09 14959.7 CO 93.8961 0.54343 7510.97 0.45657 558.633 571.42 I I I 8:59 651.2 656.49 615.51 641.067 102.684 237.879 653.143 170.282 0.97831 166.589 0.02169 614.554 520.63 | N £ a u N VO H Q Q£ o Q W 13 •Q o GA s § GA e* u II o 'S £ "re T“ .Q “O '© T“ 5 < CM < CO <1 CM JO t: + + + ! (0 © (0 time trip art 3 re Average Var[data]n Predict d in o IO o o 1 1 8:33 N. o o 630 396900 396900 396900 396900 630 1 d to CO O 3 0 o> o IO CO 3 0 I I 8:36 641.95 665.15 785.78 697.627 CO 1054.73 7771.01 2077.31 1038.66 707.89 816.42 | Tf 3 0 CM in CO 8:39 3 0 671.25 689.29 602.43 654.323 286.512 1222.67 2692.92 754.59 0.70383 531.104 0.29617 665.807 S9899 3 0 CD M* CO CD d CM 00 8:43 CM o 673.73 315.57 552.65 d ■sr 56206.9 13456 35433.6 0.50372 17848.6 579.617 501.51 | d o 00 CO CM CM I 8:59 CM nr 568.25 638.57 533.32 580.047 139.161 3424.98 2183.38 1782.07 0.91678 1633.76 530.673 632.92 | U s< •s r-~ * Q t-2 N •p* 02 o w H "3 — O GA s U A GA V II es s I £ '35 n> < CO s + r + + re re E © t trip art1 Average Varfdatajn T Predict Observed | to o SO o d o o N. I 8:33 630 396900 396900 396900 396900 630 | d 00 M- in IO 00 to d to f-. cd CO CO 8:36 00 606.12 588.43 539.63 578.06 787.364 107.537 447,45 223.725 448 I I I 8:39 3 0 566.6 657.59 8609S 595.057 809.782 3910.42 1161.22 1 2360.1 0.52263 1233.45 0.47737 507.046 678.75 | 00 IO r-- CM ■M-00 8:43 CM o 376.24 503.44 499.59 459.757 6975.03 1908.23 1586.69 4441.63 0.56096 2491.58 0.43904 423.23 I 00 00 00 d CO 8:59 CM "M* 549.87 662.18 701.45 637.833 7737.55 592.76 4047.08 4165.15 0.61512 2562.06 594.368 646.35 I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced VO - r m Table D7.28. Kalman Filter for test case RCODE = 41 0 O a . o T S ' * J < CO C CO - h *CD < . Q < > To £ 0 ■ 0 * , 2 CO CM < * * c * o t * + + + + 0 9 E 0 u a 0toCO o t <0 0totoC o t CO o t o t <0 C O b P 03 0 0 M £ 9 9 w <0 re > 5 0 o c CO CO 0 < T— . N 0 0 0 0 0 CO o c 0 O CO o t 5 0 CO O 0 0 o t CO 5 0 CO 0 0 o c 5 0 co 5 0 0 0 0 o> CO o> O 5 0 CO CO CO CO f T CO CO Tb CO 0 CM CO -’ b Tb CO 5 0 CO o t o c 00 T“ CO Tb CO d c bb#CO b-# Tb CM 0 OC M CO CO Tb 0 0 0 1 Oto t CO o t M0 CM 0 - b Oh- h CO CM o t 0 0 ) 0 CO o t CO o t 5 0 CO LO o c Tb p o t "s M" OCO CO CM ■O’ CO - b 0 0 o c - b T“ f T - b d 0 < 0 < a> CO 5 0 T“ 0 0 0 0 - b 5 0 CO 0 1 - T p CO CO CO 5 0 - b 5 0 CO CO CO 5 0 5 0 CM CO - d b 0 d CO CO •'S’ CM LO 5 0 0 - T - b V 0 d c T— 5 0 CO 0 1 o t 0 d 0 0 d i d h-* CM CO o t 0 CO - b CO - h o t CM o t co o t r 's -0; 5 0 5 0 Tb ^b d CO OCO CO 0 to CM 0 MCO ’M’ CM CM 0 < CO MCM CM M" CO d CO o t CM CO CO CO ■M- - b 0 T~ - b ^b CO 0 0 CO -’ b 0 0 o c - b - b CO - b CO 0 co - b Tb 0 TT o t CO p 5 0 0 0 1 p 5 0 CO o c o t 5 0 5 0 CD 0 o t ■*— p CM O+ CO d i co - b CM (O 09 to CM Tj- o t OC CO CO CO CO d 5 0 o t p 5 0 5 0 o t d 5 0 Tb - b Is- CM o t 05 CO 0 5 0 d c 0 - b o t o t to o t 5 0 p - b CM - b CO 5 0 o t b— - b * u T T 4 0 . H w Pi w mm Q N V O 0 Q w * te m 3 6 c i f 3 u 0 3 6 QJ to to 0 V II 1, s 3 > o T a > od To CO 0 . To s T < < to < 1 5 o * < 2 C ‘ O0 CO CM ++ 353 t t t c 0 0 r t + £ 9 0 CO 9 £ $ 9 a <0 CO <0 c MCO CM > o> CO 03 9 <0 a E 0 d d 0 < CO 0 0 0 0 0 d CO co 0 o c 0 1 o t 5 0 CO 5 0 o t o c 5 0 CO 5 0 0 co 5 0 0 < 5 0 0 0 CO 5 0 CO 5 0 co o c - > - b 0 0 0 cd CO d 0 0 d T~ CO T~ CO 5 0 5 0 CO o t 5 0 b^ ,-1 to 66 to CO O0 CO 0 0 CM CM 0 CO - b o t CO CO CO CO CM CO CO - b to CO 5 0 CM d i 0 o t ) 0 CO CO CO 5 0 CO M’ T“ 'M’ CO cd CO Tb o t o t CM 5 0 CM CO CO T— 5 0 5 0 o t CO - b 5 0 CO 5 0 5 0 CO o t T~ 5 0 CO d i 0 CO CM 0 0 1 M to o t ■ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced m VO ov Table D7.34. Kalman Filter for test case RCODE = 86 £ £ '© 'w 'S < r- < N CO N < CO r + r + + 1 time (trip . Jn CO (0 I Average Var[data]n I Predict 1 Observed | o SO o o o 8:33 ■Y~- N. 630 396900 396900 396900 396900 0.5 630 o t CM c\i 00 o to CO O LO o> CM CO o CO CO CO in 00 o 8:36 570.04 613.06 587.27 526.091 8.14151 464.715 608.64 543.67 CO m CD o> CO CO CM 00 )o O) I 8:39 658.73 471.33 510.89 5723.43 1302.73 9105.37 0.5063 4610.04 0.493701 527.07 572.92 I CO CO CM I I 8:43 00 449.86 598.27 627.48 42778.6 31548.9 853.224 37163.8 0.5292 19667.1 0.470799 586.34 723.62 CM 00 8:59 CM 612.37 579.89 531.71 574.657 1422.3 27.3878 724.842 0.96567 699.961 0.034325 538.30 587.31 If) O s <*1 ’w PS V Q ta 53 a H Q a S V Ct 4> a! im 9 1 '5 ¥ M < <3 n CO CM < t: + I time trip (0 <0 art1 Average Var[data]n a(k+1) Predict Observed | ! SO o 0 9 I I 8:33 N. o o o 630 396900 396900 396900 396900 630 I d CO o d ID o t i-^ 8:36 CO 581.56 476.12 1615.52 557.733 567.71 6660.74 3339.3 3614.22 622.76 711.86 8:39 o> 522.39 541.33 629.14 I564.287 1755.33 527.009 4205.95 1141.17 0.72095 822.724 0.279052 652.22 494.04 I CM o I 8:43 o> d CM CO 321.6 563.06 458.297 1019.74 18686 10975.4 9852.86 0.52004 5123.87 0.479961 529.93 465.82 I CO CM CO CO I 8:59 CM ■ NaiVe Model Model . 1147.79 594.2325 0.060654 351.5722 518.0104 0.100247 605.9335 0.035495 499.6859 0.093843 644.2334 0.093582 428.5672 0.014912 643.1271 0.032888 432.0833 KF Model 550.12 528.48 520.16 594.37 517.36 614.55 589.355 587.2 Model Model , 590.09 593.81 557.03 594.16 610.78 530.67 400.0117 0.034981 604.26 574.94 508.2925 607.15 595.24598.06 605.87 600.62 590.97 606.61 567.54 573.0316 615.58 623.57 856.9015 600.44 Developed Table D7.36: Prediction Error the ofTablePredictors D7.36: Prediction 572 587 606.63 538.3 620651 597.99 599.44 521 633 586 646 562 547 576 576 585 667 667 time Actual Running 6 11 17 37 26 62 91 96 80 86 72 20 41 44 49 RCODE Note: dl, d2, d3: The absolute relative error (%) ofthe Developed Model, the KalmanFilter and theNaive predictors, respectively Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 371 6 6 7 Boarders 0 0.025 0.025 0.019963 0.033386 0.010671 Observed Boarders Observed Boarders 0.011821 Observed 0.009478 0.004996 0.020503 0.012183 0.016129 0.0125 0.024886 0.009472 0.011988 Predicted 0.5 0.5 0.025 6.84E-06 0.435011 0.011907 0.025819 0.5 a(k+1) 1.97E-05 0.010662 0.018852 0.388944 0.349429 0 e(k+1) a(k+1) Predict e(k+1) a(k+1) Predicted 1.48E-05 0.339801 0.009856 3.59E-06 0.525313 0.5 0.0125 0 0.525313 0.5 0.0125 0 0 e(k+1) 8.37E-06 0.5 0.5 0.5 a(k+D 9(k+1) 0.5 0.5 0.525313 0.5 fl(k+1) 0.611056E-05 1.11 0.650571 1.03E-05 0.894747 9.2E-06 0.105253 1.050625 E-05 1.11 0.999979E-05 1.11 2.12E-05 3.73E-05 0.564989E-05 2.11 1.050625 2.24E-05 0.660199 0.000152 0.505624E-05 7,71 0.494176 0.008189 Var[data]n 1.81E-05 0.000625 1.025 0.025 0.000625 0.025 0.000625 0.5 0 0.005687 3.59E-06 0.999993 0.007334 1.025 1.050625 0.025 0 0 1.025 art1 Average 0.009478 0.009689 0.006331 0 0.028726 0.025039 0,010662 0.009307 1.04E-05 0.99998 1.04E-05 Table D7.37. Kalman Filter for test RCODE Filter case Table -D7.37. Kalman 6 Table D7.38. Kalman Filter for test RCODED7.38. = Filter case Table Kalman 11 Table D7.39. Kalman Filter for test case RCODE Filter D7.39. = Table Kalman 17 0 0 0 art2 art2 art1 Average Var[data]n 0.01333 0.005289 0.013626 0.003791 0.007384 0 0 0 0.019355 0 art3 0.003791 0 0.027036 0.015773 0.009187 0.008209 0.011056 1.29E-05 17 19 18 0 24 0.00192 24 0.013165 0.005145 0.00164 0.00665 20 0.005271 0.028798 0.005765 0.013278 trip trip art3 17 18 0 20 24 trip art3 art2 art1 Average Var[data]n 8:33 17 0 0 0 8:398:43 19 20 0.005376 0.011631 0.022259 0.004996 8:39 8:59 8:33 8:368:43 18 0 0 8:59 8:36 time time 8:33 8:36 8:59 8:398:43 19 0.011776 0.005482 time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 372 6 5 10 0 0 0.025 0.025 0.025 0.01575 0.02497 0.025921 0.016351 0.007366 0.030716 Observed Boarders Observed Boarders Observed Boarders 0.0125 Predict 0.00668 0.027982 0.009559 0.017019 0.008149 0.0158 0.013933 0.021151 0.021433 0.016883 0.010696 0.5 0.5 0.5 a(k+1) Predict 1.53E-06 2.27E-05 0.249708 0.198414 0.448749 0.496212 0 0 0.5 0 e(k+1) e(k+1) a(k+1) 1.63E-05 0.465967 4.46E-06 3.08E-06 0.103074 7.75E-06 2.65E-05 8.02E-07 0,525313 0.5 0.0125 0 0.525313 0.525313 0.5 0.0125 0.5 0.5 0.5 0.5 0.5 0.5 g(k+D g(k+1) e(k+1) a(k+1) Predict 0.750292 0.551251 0.534033 1.050625 1.050625 1.19E-05 0.999977 1.19E-05 1.41 E-05 1.41 1.050625 5.94E-06 3.05E-05 0.000625 5.26E-053.44E-06 0.503788 0.896926 0.000625 2.55E-06 0.801586 2.04E-06 Var[data]n g(k+1) 1.025 1.025 1.025 0.025 0.025 0.025 0.000625 Average 0.008277E-06 3.21 0.999994E-06 3.21 E-06 6.11 0.011619 0.011232 8.02E-07 0.999998 Kalman Filter for test case RCODE == 26 for test Filter RCODE== case Kalman Kalman Filter for test case RCODE == 37 for testFilter RCODE case == Kalman 0 0 0 art1 0.030161 0.026282 0.021151 0.018584 0.007128 0.010143 0.011619 Table D7.40. Kalman Filter for test Filter case RCODE = D7.40. 20 Table Kalman Table D7.41. Table Table Table D7.42. 0 0 0 0 0 0 0 art2 art1 Average Var[data]n art 2 art art2 art1 Average Var[data]n 0.022867 0.014097 0.022182 0.011964 0.019298 0.010571 0.007254 0.010645 0.010696 0 00 0 0 0 art3 art3 art3 0.01273 0.019963 0.009139 0.017871 0.015658 0.021263 0.007945 0.006005 0.011738 0.012354 17 17 1819 0.020503 0 0 19 0.005959 18 18 20 0.025819 19 20 0.023748 24 20 24 24 0.008799 0.00783 0.00375 0.006793 trip trip trip 8:33 8:43 8:59 3:33 8:36 8:36 8:39 8:59 8:33 17 8:43 8:36 8:39 8:39 8:43 8:59 time time time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 373 8 11 11 Boarders 0 0.025 0.007453 0.010491 0.013894 0.010971 0.014896 0.006208 Observed Boarders Observed Observed Boarders Predict 0.01722 0.014186 0.018551 0.012378 0.008281 0.012947 0.010676 0.018601 0.017688 0.5 0.025 0.5 0.025 a(k+1) 0.281579 0.443661 0.013359 0.395638 0 0 0.5 e(k+1) a(k+1) Predict e(k+1) 3.7E-05 0.470891 0.019089 1.71E-05 1.06E-05 2.03E-05 8.65 E-06 8.65 1.65E-05 0.525313 0.5 0.0125 0 0.525313 0.5 0.0125 0 0.525313 0.5 0.0125 0.5 0.5 0.50.5 0 g(k+1) e(k+1) a(k+1) Predict g(k+D a(k+n 0.99998 0.529109 0.688638 3.14E-05 0.311362 0.718421 1.050625 1.050625 8.65E-066.99E-05 0.999984 2.02E-05 0.604362 1.22E-05 0.000625 0.5 0.000625 2.52E-05 0.999952 2.52E-05 4.79E-05 0.020521 2.42E-05 0.600668 1.45E-05 0.399332 Var[data]n 1.025 1.025 1.025 1.050625 0.025 0.025 0.012467 2.39E-05 0.010827 1.06E-05 0.012709 0 0.025 0.000625 0.5 0 0 0 0.028708 0.013279 0.015101 0.019544 0.011669 0.017221 0.020522 Table D7.43. Kalman Filter for test case RCODE = Filter41 RCODE test for Kalman Table case D7.43. Table D7.44. Kalman Filter for test case RCODE 44 = Filter RCODE test for Kalman Table case D7.44. Table D7.45. Kalman Filter for test case RCODE 49 RCODE = Filter test for Kalman case Table D7.45. 0 00 0 0 art2 artl Average art2 artl Average Var[data]n art2 artl Average Var[data]n 0.006657 0.012378 0.010411 0.006447 0.009325 0.022195 0.011069 4.55E-05 0.004787 0.006981 0.011553 0.021491 0.013499 0.011953 0.024276 0.019062 0.015979 9.92E-05 0.556339 5.52E-05 0 0 0 0 0 0 0 art3 art3 0.012199 0.002334 0.008796 0.001686 0.010676 0.007453 0.005651 17 17 18 19 18 0 0 18 19 0.008281 19 20 20 24 0.015854 20 0.004598 24 24 trip art3 trip trip 8:33 17 8:36 8:39 8:43 8:59 8:36 8:33 8:36 8:33 8:59 8:39 8:43 8:39 8:43 8:59 time time time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 374 7 4 4 Boarders Boarders 0 0 0.025 0.025 0.025 0.00287 0.010486 0.013777 Observed 0.0125 0 0.0125 0.0125 Predict Observed 0.01167 0.005818 0.014023 0.007805 0.009604 0.012921 0.005737 0.010425 0.010422 0,018433 0.5 0.5 0.5 a(k+1) Predict Observed Boarders a(k+1) Predict 0.41494 0,018428 0.012969 1.02E-05 0.375445 0.006315 0.486958 0.220337 0.012001 0.013673 0 0 e(k+1) a(k+1) e(k+1) e(k+1) 1.01 E-05 1.01 0.434693 1.94E-06 3.69E-06 5.36E-06 8.74E-05 0.525313 0.5 0.525313 0.5 0.5 0.5 0.525313 0.5 0.5 0.5 0.5 g(k+i) g(k+i) g(k+D 0.99999 0.58506 8.95E-05 0.565307 0.526458 9.12E-06 0.473542 0.999996 0.513042 0.779663 2.75E-05 1.050625 1.94E-06 1.050625 0.000625 0.5 0 0.000625 6.27E-05 0.999881 6.27E-05 0.000119 Var[data]n Var[data]n Var[data]n 1.025 1.050625 1.025 1.025 0.025 0.000625 0.025 0.025 0.018658 1.79E-05 0.012659 1.52E-05 0.624555 9.5E-06 0.011615 0.016948 0.016931 0.000153 0.016881 3.53E-05 0 0 artl Average 0.01487 0.01167 0.005419 0.014563 0.007176 1.73E-05 0.018435 0.021727 0.011727 Table D7.46. Kalman Filter for test case RCODE RCODE = Kalman test D7.46. 62 for Filter case Table Table D7.47. RCODE Kalman = case D7.47. test for 72 Filter Table Table D7.48. Kalman =RCODE test Filter case D7.48. 86 for Table 0 0 0 0 0 0 art2 art2 artl Average 0.010196 0.005403 0.030089 0.008458 0.014407 0.00017 0 0 0 0.012978 0.004674 0.002397 0.026668 17 17 0 17 18 24 0.014108 0.024808 trip art3 trip art3 art2 artl Average 8:33 8:36 18 0 0 0 8:59 24 0.014829 0.01773 8:39 19 0.01388 0.016593 0.009604 0.013359 5.36E-06 8:43 20 0.024329 0.016775 8:33 8:43 20 0.001562 time trip art3 8:368:39 188:59 19 24 0 0 0 8:33 8:39 19 0.008319 0.024091 8:43 20 time 8:36 8:59 time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 375 8 8 Boarders 0 0.025 0.018217 0.017987 Observed 0.0125 0.0125 0 Predict Predict Observed Boarders 0.015659 0.006982 0.014877 0.008292 0.007947 0.013039 0.008513 0.012377 0.023614 0.5 a(k+1) 1.99E-05 0.007357 0.013478 0.474441 0.450247 e(k+1) a(k+1) e(k+1) 6.4E-06 1.05E-05 0.525313 0.525313 0.5 0.000201 0.5 0.5 g(k+i) 9(k+1) 0.992643 1.49E-06 0.525559 1.5E-06 1.050625 1.16E-05 0.549753 1.050625 0.000625 0.5 0 0.5 0.000625 0.5 0 0.5 0.025 4.12E-05 0.999922 4.12E-05 7.85E-05 Var[data]n 1.025 1.025 0.025 0.01442 1.05E-05 0.99998 Average Var[data]n 0.020252 3.17E-06 0.811191 2.57E-06 0.188809 0.015165 0 0 0.025 0 0 Table D7.50. Kalman Filter for test case RCODE test Kalman = for case D7.50. 96 Filter Table Table D7.349. Kalman Filter for test case RCODE test = case Kalman for 91 D7.349. Filter Table 0 0 0 0 art2 artl Average 0.022229 0.016715 0.022168 0.007947 0.01358 0.059079 0.0071040.016898 0.013403 0.03158 0.000383 0.010347 0.016926 0.011268 0 0 0 art3 0.021813 0.006532 0.028558 17 0 18 19 0.010626 18 24 8:43 20 8:39 19 0.010626 0.016973 0.015659 8:33 17 8:33 8:36 8:39 8:36 8:43 20 8:59 24 0.015194 8:59 time trip time trip art3 art2 artl Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. d3 0.9091 0.1667 0.1250 0.5833 2.4000 d2 0.53850.2308 0.6923 0.5385 0.3333 0.3750 0.3333 0.8333 0.4000 0.0000 0.5000 0.7000 0.3333 0.1429 0.7143 0.25000.3000 0.7500 0.33330.2500 0.8333 0.6250 0.2727 0.4545 0.3846 0.4615 0.1667 0.83330.1667 0.5000 0.1000 0.1250 0.0000 BBMBj 1 1 4 6 3 0.1250 2 0.1250 19 0.3333 Naive Model 5 3 0.2000 6 8 5 0.1667 7 34 6 21 8 2 0.0000 4 4 11 7 kf Model 7 10 7 7 7 7 5 0.4000 7 10 7 6 8 7 11 9 Model Developed 7 7 6 7 5 8 6 6 11 8 13 10 8 12 8 8 Actual Boarding Passengers Table D7.51: Results Boarding Predictors 3 of the Prediction Passenger D7.51: Table 6 8 17 11 10 9 37 6272 6 96 2026 13 49 91 8086 8 41 44 6 RCODE Note: dl, d2, d3: The absolute relative error (%) ofthe Developed Model, the KalmanFilter and the Naive predictors, respectively Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 9 9 11 12 11 12 13 0.025 0.025 Observed 15 20 13 18 11 12.0416 15 10.07937 11.60054 9.507692 9 0.5 0.025 0.5 0.5 0.5 0.0125 9 0.5 0.0125 a(k+1) Predict Observed 0.438283 0.439895 0.460315 a(k+1) Predict Observed a(k+1) Predict 0.144416 0 0.03125 0.5 0.0125 0.03125 1.589072 e(k+1) e(k+1) 2.302652 e(k+1) 0.5 0.5 0.03125 0.5 g(k+D 0.561717 1.123434 0.539685 0.590172 2.131178 0.409828 g(k+i) 0(k+1) 0.855584 0.237662 0.600544 1.367906 0.399456 2 0.0625 0.000625 0.000625 0.5 0 2.277778 0.503407 1.146648 0.496593 9 4.111111 0.560105 2.944444 2.277778 0.277778 Var[data]n 17 10 1 0.507692 0.507692 0.492308 0.25 0.0625 15.66667 1.111111 0.506934 0.56326 0.493066 16.33333 10.33333 3.611111 12.33333 Average Var[data]n 0 0 0.25 17 14 artl Average Var[data]n Table D7.53. Kalman Filter for test case RCODE for test RCODE =6 Filter case Table D7.53. Kalman Table D7.52. Kalman Filter for test case RCODE RCODE for test Table= Filter case 1 D7.52. Kalman Table D7.54. Kalman Filter for test case RCODE for test = TableFilter RCODE case 11 D7.54. Kalman 7 11 9.333333 0 0 19 15 18 art2 artl art2 art2 artl Average 0 0 0 0.025 0 0 14 11 12 11 10 10 10.33333 art3 art3 17 17 0 0 0 0.025 0.000625 0.5 0 17 0 0 0 0.025 18 19 15 17 15 18 0 0 0 0.25 0.0625 0.5 19 11 9 10 18 19 8 11 9 9.333333 20 17 24 14 24 9 8 20 10 24 20 trip art3 8:33 8:33 8:36 8:39 8:43 8:36 8:33 8:59 8:39 8:43 8:59 8:36 8:39 8:59 8:43 time time trip time trip Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 18 8 15 17 0.025 0.025 0.025 Observed 11 9 17 19 19 17 22 15 0.0125 9 0.0125 8 0.0125 19.14278 Predict Observed Predict Observed 9.526627 11 Predict 0.5 0.5 0.5 0.5 0.5 0.4361920.470862 12.69143 0.4956440.482152 11.53049 15 0 0 0 2.47619 0.380952 e(k+1) a(k+1) e(k+1) a(k+1) 0.146285 0.473373 e(k+1) a(k+1) 0.896633 0.281904 6.271712 4.552387 0.344456 1 0 0 0.5 2.5 0.5 0.5 0.5 0.03125 0.5 0.5 0.03125 0.5 g(k+D 0.526627 g(k+D g(k+i) 0.504356 0.563808 0.529138 1.205258 0.655544 5 0 4 0.619048 0.5 0.0625 0.5 0.03125 0.0625 0.0625 0.000625 0.5 1.777778 0.000625 12.11111 0.517848 0.277778 6.944444 Var[data]n Var[data]n Var[data]n 19 17 13 0.25 0.25 0.25 0.025 0.025 10.33333 2.277778 12.33333 12.66667 0 0 17 17 0 0 14 10 15 17 23 16.33333 artl Average Table D7.55. Kalman Filter = RCODE case for test Kalman D7.55. Table 14 Table D7.56. Kalman case RCODE Filter = for test Kalman D7.56. Table 17 Table D7.57. Kalman Filtercase =RCODE for test Kalman 23 Table D7.57. 0 0 15 15 15 22 0 0 0.025 0.000625 0 0 0 11 13 14 art2 artl Average art2 art2 artl Average 15 18 0 9 12 11 10 10 10.33333 11 12 art3 art3 art3 17 0 1819 0 20 24 19 15 17 0 0 17 0 19 18 19 18 0 20 12 24 9 12 24 20 trip trip trip 8:33 8:36 8:39 8:43 8:59 8:33 8:33 8:39 8:36 8:59 8:39 8:43 8:36 8:59 8:43 time time time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 8 8 19 0.025 0.025 0.025 Observed Observed 17 17 16 17 10 15 13 17.5 15 0.0125 8 0.0125 8 0.0125 8 11.51362 19 Predict Observed 0 0.5 0.5 0.5 0.5 a(k+1) Predict 0.498054 0.294985 18.29498 0.369684 a(k+1) 0.417895 9.746315 0 0.5 0 12.19187 10.74349 0.36803 2.007782 0.910457 e(k+1) 8.230027 0.499525 8.500475 e(k+1) 0.513514 0.486486 e(k+1) a(k+1) Predict 1 0 0.5 0.5 0.03125 0.5 0.03125 0.5 g(k+D g(k+i) 0.582105 g(k+i) 0.630316 0.501946 0.705015 1.018355 0 4 17 0.63197 0.0625 0.0625 0.5 0.03125 0.0625 1.444444 1.444444 16.44444 0.500475 0.000625 0.5 0 0.5 0.000625 0.5 0.000625 20.94444 Var[data]n Var[data]n Var[data]n 16 1 0.513514 0.25 0.25 0.025 18.66667 0.111111 0.5 0.055556 15.33333 Average 0 0 0 14 16 11 12.66667 artl Table D7.60. test 37 Kalman Filter for RCODE = D7.60. case Table Table D7.58. Kalman Filter for test case RCODE test for RCODE = 26 Kalman Filter case D7.58. Table Table D7.59. test Filter for RCODE Kalman = 30 D7.59. case Table 0 0 0.025 0 0 0.25 7 9 10.33333 0 9 17 17 17 17 17 1319 15 18 19.33333 15 art2 artl Average art2 artl Average 0 0 0 0.025 0 0 17 15 19 19 18 17 9 10 12 art3 art3 art3 art2 17 0 0 17 0 19 18 18 0 19 17 18 0 19 15 24 20 21 20 24 15 20 18 24 trip trip trip 8:33 8:33 8:33 17 8:39 8:36 8:59 8:43 8:36 8:43 8:59 8:39 8:43 8:36 8:59 8:39 time time time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 9 7 9 13 0.025 0.025 0.025 Observed Observed Observed 9 8 9 8 15 15 0.0125 0.0125 9 Predict 11.0155 17.27679 15 6.989298 0 9 0.5 0.5 0.5 a(k+1) 0.360635 10.27873 9 0.496124 0.355207 0.494649 0.239587 0 e(k+1) 0.03125 0.5 0.5555560.666667 0.5 0.4 8 9 e(k+1) a(k+1) Predict 0.03125 0.5 1.007752 3.623393 0.465401 2.865749 0.729951 e(k+1) a(k+1) Predict 0.603845 1 0 0.5 g(k+i) 0.5 0 0.5 0.03125 0.5 0.0125 0.5 S(k+1) 0.505351 0.639365 9(k+1) 0.760413 0.211226 0.534599 2 0.503876 1.111111 0.6 1.111111 Var[data]n 0.0625 1.444444 0.000625 0.5 0.000625 0.944444 0.277778 6.777778 4.444444 0.644793 Var[data]n 0.025 0.000625 0.5 0 15 Average 7.666667 0.25 0.025 0.025 7.333333 17.66667 8.666667 7 0 9 9.666667 art1 0 6 0 0 0.25 0.0625 9 11 9.333333 15 16.33333 21 art1 Average 9 09 0 9 0.25 9 0.0625 0.5 0 Table D7.61. Kalman Filter for test case RCODE 41 RCODE = test Filter for case D7.61. Kalman Table Table D7.62. Kalman Filter for test case RCODE RCODE for 44 test = Filter D7.62. Kalman case Table Table D7.63. Kalman Filter for test case RCODE RCODE for test 55 = D7.63. Filter Kalman Table case 0 0 9 0 0 15 18 art2 art1 Average Var[data]n 9 11 0 7 0 0 0 0 17 15 13 14 art3 art2 art3 17 0 0 19 9 18 20 24 7 trip art3 art2 17 0 19 18 19 18 20 9 8 24 19 24 9 8 20 trip trip 8:33 17 8:36 8:39 8:43 8:59 8:33 8:59 8:36 8:39 8:43 time 8:33 8:43 8:36 8:39 8:59 time time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 15 0.025 Observed 17 0.0125 9 Predict 17.69581 15 0.6667 0.1333 0.5455 0.2941 0.1176 0.1333 0.2105 0.2222 0.0000 0.1538 0.5 0.496124 13.03101 0.431359 d2 d3 0.3846 0.16670.1667 0.0909 0.1364 0.1818 0.11110.1818 0.0000 0.1111 0.1250 0.1333 0.0667 0.1176 e(k+1) a(k+1) d1 0.5 0 0.5 0.2222 0.09090.1579 0.0833 0.0556 0.13330.0000 0.0526 0.0000 0.0000 0.0588 0.1579 g(k+i) 2 0.568641 1.137282 2 0.503876 1.007752 7 0.0000 0.0000 7 0.0000 9 10 0.2222 17 0.0000 19 15 14 0.0588 13 0.2308 11 22 0.0667 Naive Model 0.000625 0.944444 0.673952 0.63651 0.326048 Var[data]n 8 9 9 9 0.0000 13 15 11 17 13 10 8 16 17 KF 16 0.25 0.0625 0.5 0.03125 Model 0.025 Average 17.33333 0 19 18 17 15 11 16 19 11 18 16 15 Model Developed Table D7.64. Kalman Filter for test case = RCODE test case for Filter 59 Table D7.64. Kalman 17 14 15 9 11 10 15 11 15 17 18 19 16 1517 15 16 15 Actual Table D7.65: 3 Results Predictors Boardingthe Prediction of D7.65: Passenger Table number of number De-boarders 13 16 1 6 11 10 11 14 9 17 37 8 8 20 23 30 1755 59 16 4149 8 8 8 8 26 13 44 9 9 RCODE 17 0 0 18 0 0 0 19 20 24 16 Note: dl, d2, d3: The absolute relative error (%) ofthe Developed Model, the KalmanFilter andthe Naivepredictors, respectively 8:33 8:36 8:43 8:59 8:39 time trip art3 art2 art1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix D8 1. General This appendix covers the mathematical concepts of the Tukey test used for the purpose of evaluating the models’ performance. As discussed in chapter 6, section 6.4.3.6 and section 6.4.1, MAPE is the most suitable criterion to be compared so that the performance of the models can be ranked. 2. Tukey Test Procedure The results presented in chapter 6 were obtained by following the steps suggested by Netter et al., (1985). Step 1: Statistical hypothesis Ho: MAPE N= MAPE K= MAPE P Ha: MAPE n> MAPE K > MAPE k or MAPE K> MAPE P > MAPE P Where N = Naive model, K= Kalman Filter Model, P = Developed model Let i = The number of models being compared; i=3 j = the number of elements of the sample test Select level of significance a =0.05, then select suitable Qa, /„ uj-i) from the studentized range distribution. Step 2: Determine range w = Qa JM SE / j where MSE = mean square error within group. 382 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 383 Step 3: List the sample’s MAPE in the increasing order and make pairwise comparisons. If the difference is smaller than the predetermined range, the null hypothesis cannot be rejected, or there is no statically significant difference between the pairwise models at a = 0.05. Such a difference is underlined. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix D9 1. General This appendix covers the predicted arrival times of 15 full trips at three selected stops. These are Holmwood, Gladstone, and Rideau stops. 2. Prediction Results The results are tabulated in Tables D9.l-D9.45. The left hand side part of the tables shows the predicted running times between stops and dwell times at the stops by applying developed running time and dwell time prediction modules, respectively. The right hand side part of the tables presents the predicted arrival time obtained from the general equations 3.4 and 3.5 (chapter 3). The updated predictions followed the procedure presented in chapter 3, section 3.3.4. The following legends are used in Tables D9.1 to D9.45 T6-R1-54: Running time from stop 1 to stop 54, trip 6; T6-R54-3 : Running time from stop 54 to stop 3, trip 6 Boar. Number of boarding Passenger at a stop Ali: Number of alighting passenger at a stop Dwell: Dwell time Arv.Holwood: Arival time at Holmwood stop 384 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (number): The number showing the times the prediction is updated. Error: Prediction Error, the difference between the predicted arrival time at last stop to that of schedule time. %. Relative Prediction Error Numbers in the colored boxes are the predicted values Numbers in non-colored boxes are the actual values Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 386 R ideau 164342 1588.30 1615.50 1239.50 1600.11 1626.49 1255.62 Rideau Arv. Arv. 1014 1701.50 1024.55 670 Gladstone Gladstone 1024.13 959.86 982.69 Arv. Arv 556.61 604.5 Holmwood Holmwood Arv. Arv. T6-R3-5 3(3) T6-R3-5 T6-R3-53(2) T6-R3-53(1) T6-R3-53I2) T6-R3-53(3) T6-R3-53(1) 569.5 630.5 614.92 673 1214.5 605.42 513.47 14.5 dw ell3 7 583.95 dwell3 "iS*:: 603.94 10 32 3 2 4 17 585.36 kf T6-R54-3(2) T6-R54-3(1) boar3 all3 T6-R54-3(2) 397.5 407.63 12 3 dwell 54 39.5 dwell54 T6-R54-3H) boar3 all3 /3 4 ’5 " :': .i i i;'l! all. 54 all. ali. 54 Route 1-Direction: Downtown 8:59 a.m. bus trips. The Developed Model 13 boar54 boar54 ; i i ? » ':, '. : : 630.5 604.5 2 t6-R1-54 513.47 457.22 t6-R1-54 556.61 Table D9.1- D9.15. Real-time Bus Arrival Prediction at Stops: Holmwood, Gladstone, and Rideau. 1st 1st 2nd Update 2nd Update last last 17 12 dcode d co de Table D9.2 Table D9.1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced Table D9.3 Arv. Arv. Arv. *<5 15 g £2 E *n M* P) CM 0 CM « •Q <2 £2 5 v> CO 1 in co « 3 •Q •s tf> i Arv. Arv. 1 g on •o P s E q? an Mi "5 N*. <2 to co S m n i Arv. Arv. p 1 g n s *Q CO S sn •Q i s g *» to *2 CO CD ft m *8 ■O 1 o o <8 o <0 o • • • Holmwood Gladstone Arv. R ideau <;«> i::CO iiCIi i;0 M iO' * 8 8 g V- OJ T- sn s S in CO CO s o 09 co T- w CM o> CM in o> 00 in o h- g 3 s so co E 1 co t f M ;;in. i t t- N 00 a> 10 r- s *D 1 1489.93 I g co 3 CM s m CM CO o> in CO O h* 5 W) 1529.87 | eo in h. CO o> >- C> co K § Oi N» 7500.37 | Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced Table D9.6 Arv. Arv. Arv. S E I 2 £ i 2 £ "O ? •Q £ o « CO P i A r v . Arv. Arv. te S 15 CO i E 3 •Q 2 * i o c £ co p 3 o: I IO 8 X 1626.50 I Os H 3 Q 0 0 a a> Arv. Arv. A r v . i E CO o c £ s CO 15 p p x * to P Table D9.9 Arv. Arv. Arv. i Q £ § £ % 3 ■Q E S *o co W) ■o to «Q 1 ■2 o <0 1 O <9 dcode Update <9 ali3 Holmwood Gladstone Rideau w MCM CM i p i ;§* ?oq* 4oP .CM. .© 1 :;l i iHDi § .5 CM rt ▼*: in CO CO o M CD *» «* 1582.69 I M S CD CM CM CD 3 E 8 So E t co d o> 00 CD CM CM in CO CD m oo h*. 00 m <5 ■fc 1613.80 I CM £ 5 ? IO CO CO CM ■ m in in T? ■2 w to I 1597.45 I N O) o> CD CO CM CM O 1676.00 | 2 O H q O es a» n Arv. Arv. Arv. ~n I E s P -O CO •2 co in E K <*> i u i a; *Q ■8 *8 >*• 5 $ I o 0 l O <9 4) I i Holmwood Gladstone Rideau • «& ':;T“ :iOOi , ;:iK m « o y s CM CM CD » MC m CM CD CO i n CO 00 CM N CM 00 CO >» to »>' .. CM ■O’CM in CD E co § s IO E a: j co p© iiiCr O) 3 m CO 00 CO tn CO § s <5 ■o 1581.21 I CM CM E CD £ lo JO CO CO OO D * me i v :1D: Oi r- -2 w> 1590.66 I o> O) IS.* V* in CO K m U) 1590.50 1 2 O H Q es a> n Arv. Arv 4 rv . ~n i £2 22 4 © ■io -Q E 8 ) V E 5 i i m QCO -Q a: 8 * ■§ ■8 M- CO 5 1 O <9 <9 O o 2 (9 • i Holmwood Gladstone Rideau :;CD ::o © So D> 3 N co s CD » m T~ 4 m !n o> a © I 1274.42 | 1891.46 I o> in © r-. © o in 5 E <0 2 •o 5 £ co .'.Tf "Ss 8 CO co K of CD A CO co in tn CD 5 *D I 1336.06 | o> to P- CO CM V) T6-R3-53(3) \ 00 o t © © © •2 U) 2017.58 | to © © >- *0* © c o sso o I to to to to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced O o m n Table D9.12 1...... Arv. Arv. Arv. £2 5 1 °? £ 4 ■o -Q 75 c Arv. Arv. Arv. 75 CO | K 1 § £ | ■Q -i S o: § *0 dcode Update • dwellS4 Holmwood Gladstone Rideau o> * CM m s CO r- N o> CO CO CM *r» CM m oo CM CO CO 3 CO !? IO v> 1793.04 | M CO CO CO CM CM IO IO CO - f CM § c £ £ CO 5 IO Oi M in T" CO CO m CO e CO oi m 1 K m £ "O 6 1783.68 | 5 CO CO o CM CM £ 11 CO co 10 « 1 -;«0' T“ CM 1 1870.02 | m CM IO i*. o> N. CM cd IO IO 1848.00 | H IB Q O t -H o V n Arv. Arv. Arv 1 > £ s Q oo IO 1 3 -Q £ CO •¥ •*- 3 CO w o <0 dcode Update ■ all54boar54 ali3 Holmwood Gladstone . Rideau OTt T CO CO CO IO 1st I 558.67 I I 14.5 I 472.99 I 14.5 I 613.75 I 558.67 | 1046.16 | 1674.41 I fx. 493 ! 9-5 I S 3 £ s £ v> co s o c 1 IO CO 2nd 399.17 I 588.96 I 901.67 I 1505.13 | £ (* o c IX. 8 393 1 i last 610.95 I ! 1523.45 I 572.5 493 895.5 1485.00 | Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced m ON Table D9.15 Arv. Arv. Arv i 5 i IS S -o JO £ o t •Q 2 • CO 2 £ ■o to **» to M- "3 o <0 0 (0 • dcode Update 1 Holmwood Gladstone . Rideau * * 1 W § :C> V « to « IO 3 0 0 b i w co CO ) O IO «*> OIO IO CO h- o T- to 1372.33 I T~ CO CO o> © o> IO IO IS 5 § W) tc 3 s CO IS co • l C -*• N C- o 00 T“ - h h- to 0ft 3 - r IO <5 1368.12 | - r s 5 V) CO M n i ? co co 0 o o oo 5 0 1338.68 I § *o <0 m oo Co m r 1397.00 I 392 1st Updated 1556 56 last 1527 13 1627.23 2st Rideau Update Arv. 1445.62 1604.57 1214.331304.00 last Rideau Arv. 1012.5 1582.00 1010.01 616.5 1076.79 Gladstone 955.11 Gladstone Arv. Arv. Holmwood 604.5 630.5 Holmwood Arv. Arv. T6-R3-53(2) T6-R3-53(3) T6-R3-S3(3) 569.5 673 547.52 583.33 544.06 577.56 574.08 480.39 T6-R3*53(1) 24 5 24 14.5 dwell3 T6-R3-53(1) 12 dwell3 14.5 32 7 543.44 all3 a/r'3 boar3 3 4 17 3 2 7 boar3 10 T6-R54-3(2) T6-R3-53(2) T6-R54-3(2) 393.51 T6-R54-3(1) 342.5 367.15 406.79 T6-R54-3(1) dwell 54 dwell dwell54 12 39.5 12 Route 1-Direction: Downtown 8:59 a.m. bus trips. Kalman Filter-based model ali. 54 ali. 13 boar5454 a!i. boar 54 2 Table D9.16 -D9.30. Real-time Bus Arrival Prediction at Stops: Holmwood, Gladstone, and Rideau. 30.5 6 604.5 2 t6-R1-54 t6-R1-54 480.39 17 508.01 ,'i 412.6 dcode 12 dcode Table D9.16 Table D9.17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced Table D9.18 Arv. Arv. Arv. 3 — ft *“ 'to £ 3 £ ft CO 1 Q ft CO IO o e to CO IO «Q ? IO i ** o <0 CD o id dcode * Holmwood Gladstone Rideau Update o ■:::0 M :--co ■Mjf to OC CO CM CO T“ IO | | 544.4 CO F:i4.5J : 636.39 | 544.40 |1 1012.80 | as ] ass s M o V* © 9.5 £ co co £ 1 *? CM ft: IO :I:CN|. ,1s* >r N Mfr T~ Mf to 3 CQ I 448.01 IO 596.97 I 1016.01 | £ CM CM CO 373.5 ft IO co :.<© v> o» 1 663.09 I /ast | o M* oo N. 534 558.5 941.5 IO Arv. Arv. Arv. 1 CO ft t f 4 -Q *<0 I- CO IO to o c IO CO 2 < o c 1 X H CO 3 ft IO CO •Q o CD CD o 1 dcode ali54 ( Holmwood Gladstone Rideau Update M ;;Jc*! ■m i m d le ■jft; d a CM o « 09 * CO CO o e s i 3 Y» o ' s CO m n t CO CO CM •o - h o *0 Mr CO tn M IO M’ a* CM CO e s s IO CO £ <¥ S CO ;:|Sr ■ tn. :.#!• A k l C o t o o s n i c IS to o e : " T h*. « k * © CM co (ft ** S I o c s CO to s u> o c r*- CM o t © 3 CM to to S CO \ last 1 © 0 0 o c o> - i CO k IO IO CO M" 09 CM . K Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced Table D9.20 < < z < > L. £ . 1 *o O tc ■O tc 3 in 1- CD nm 0 CO in CO < tC m CO T“ to ■o K JO T3 ■a co «Q aCO -a 50 © o o > © o c O O © O co CO 5 0 <0 CO $ © i U pdate i:(© 1*0 iSf i:Ot :..T" :**• m •** N © «M m C7»N « to 00 s CO Tf a> CM co 1- CO CO Is- 0 0 *«• * m in ■+>* m (4 CM 0 o> 0 0 Is- o> in CM in K CO or 5 coor co co CM <0 K- co • I** 0 * O CM o> I-. M a> CM CO K CM K CO in CO cn CM CO T- ** to V to K co o> in Or to CM to ■ '■vH :>0 0 * CM CO to to CM 0 fast | T- O O 0 0 tn O o> CO o> *» CO 0 co K. »o K m 1 K ^ Arv. Arv. Arv. •O H CO CO CO £ 1- CO tC in CO <2 *© 50 52 tc in tn 1 CO -Q ++ a 3 © © 1 0 © 0 © © © t < ■ code t Holmwood Gladstone Ride; r«n ?co ':x* HM*: M' >* X5 th- 1 CO L490.25 I m I 391.63 590.11 I 490.25 | 1 896.38 | 538.5 S'6 0 r» ,R £ 2 co in CO £ tc to ■ iilO hWBi: last to ’M O) O 659 538.5 983.5 co H Q N N 3 OS « Arv. Arv. Arv. h- o c CO 1 I *« CO ■O tc s tc CO in *© 3 52 H CO CO -a •a © I O © 0 • dcode t6-R1 -54 1 1 Holmwood Gladstone Rideau Update :|S i im ‘:i£ s I .<*»• v > m -an ;«* M> K to o> IO ♦* to CM 3 CM CO Is- CM MCM ■M CM o t i m © in CM CO 0 O m 3 £ 8 £ ? M 3 5 c-> 1 1 .*© ;*• 'id N CM S n O o> CM o t co 3 int oi in in h* in 5 v Mi in £ CO i O o t m 4 o t CO co :: Table D9.23 0 < K < 2 * l a > > > > — . > . E . "O o * "O £ 0 w £ 0 K | 2 £ - 1 *n CO TJ ■o CD £ s | M* - 1 CD n i CO T“ r ^ -Q CO 15 CO CD -Q m w> IO © © 3 5 o o c • A o © © n i © V. o o o © . k © o © a ( % • U p d a t e © IO o « IO CO h o ID * K «M 1“ CO CD CD CO m CO M CM - b CO T* COCD IO CD IO CM o IO CD CO f * ( 1 CM oi m T” 'C n i s CD 0> CO CO s § JO m CO m CO 3 JO «? • • CO o OJO o « o CO CO CM o o CO Y* CD <© CD o c T" CM K CD o MCM CM CD CO T” * n i CO 2st \ s M1 o T— ©r o j S»H o o CO 0 0 m ©■ o #*■* CO OQ> CO K last \ CO >» O m o < >» Tf H i S Q ON C3 ■5o <0 X < b 5 b 13b > > >"O >or © • . . E . OCO CO « “O -O H Q£ CO in CD s 15 CO ■o CD Of •Q mi § *3 JO ©• H CD of CO •Q co £2 *D in in o o © © o c © 3 5 © k. © o o © © © o © o i * Update ;.p i:«* SMfr ':*&■ ill l«M i:o ' :o K 5? CO t O o CM CM m ed 2 CM CD CM CD i O CMCDCDo CO M cm CM ©* o o> CM MCM CM T“ CM r*. CD s 3 K v> m JO c* v> 4 § * >» K O CMo> fk. CD CD s CO 00 5 K CM s 3 JO ■*“ COCM a lie 3 CO K m s CD © h» m last \ o> cd o © o> o to CO CM CM ON N in H Q 4> « H 15 CO "O CD Of CO n i CO © £2 I s •Q £ * •Q S 15 v> n i i o © o © dco de a T6-R54-3(1) Arv. Arv. Arv. U pdate Holmwood G ladston e Rideau "w i iie e ; jiC»' :ie e i ;pl m s 5 CO © CM eo V) T“ CO © CO h * CO CO CM CM oo © © CM CM CM CD n i o t ss S m co 3 o: 5 3 1? CO a ;;Tfi: ;;co e . o CM © o ▼- d S © M o © n i CD m o » CM e S n i co 3 CM CO CO 00 :;in 0 1 3 © * fast 1 n i e s U) o> K V) o> o © to m o Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced Table D9.26 H CD Of CO n i CO 1 3tOi aro I * 0 M i H S I 3 co N >■ R CD CM s n i CO K. in K CM CM K m CM CD i o CO ** m m T" - r T“ § i o a> n i CO M n i S $ !S 3 s m co 7 :;*0 H I ■I'fr IS ci :c « • > K G. » o A CO s CO n i s I K ** o> n i N 5 s CM J2 H m CO CD i « f :jco !;*» ;itv ■s 5? s *> i o S CO ’M* m - * CM CM o> in M- o> O) m i o m T*» - * in CD H o N 0 0 3 O «s a> n : a < X < - I T* o < CD Of CO n i n 15 CO i O! •Q 7 co - 1 CD m CO *5 73 CD £ QC CO CM -Q 1 ■o T3 +* o 2 l i E 9 o (0 o « m O O $ • 1 ® ■ E E . U p d a t e "D +* ffi O O w o c © 3 3 >*• CO Oi T“ CO 1*^ T“ CO o T* CM Oi 00 CO o» o i o co in CO CD 00 - r n i CO N CO CD V* CO T“ CM CD CO in (0 CM CM CO Mr V) CM in N 82 4 3 m co OC 3 £ 4 CO m * t:CD. •;1*S ::0 im CM «0 Table D9.29 H CO 9r CO 1- <0 tc 3 CO *2 *c5 co T3 co in CO CD £ in •a ? i 1 "© i O © I O CD CD * 1 dcode ■ A rv . A rv . A rv. U p d a te H o lm w o o d G l a d s to n e R id e a u m .Y- !!«a :-TT ;•;© •M ■.»- ;Mf Cf tt IO CO N o> o CO T" CO to o> *- s CO CM o> U) to CO IO CO in © - r CO 0» in o> § K e s m co CK s co 1 • IK: § "W •is V© ;-CO © 9 CM «n S ' s (D 5? o> m m in co * co 2 CO o> 1582.00 1372.68 1st 1513.93 1579.06 1701.50 1632.50 1372.66 1577.00 Rideau Updated Rideau Update Arv. Arv. 1012.5 1014 Gladstone 828.32 1109.50 967.44 Arv. Gladstone Arv. 630.5 Holmwood Holmwood 604.5 Arv. Arv. 465.59 T6-R3-S3(2) T6-R3-53(2) T6-R3-53(3) 673 569.5 6 20 6 558.89 436.09 801.79 529.49 550.56 T6-R3-53(1) 12 14.5 14.5 32 dwell3 24.5 dwell3 T6-R3-53(1) a//3 a//3 10 3 3 boar3 boar3 2 T6-R54-3(2) T6-R54-3(2) 397.5 342.3 T6-R3-53(3) 328.23 350.94 4 17 T6-R54-3(1) 12 dwell54 39.5 dwell 54dwell T6-R54-3(1) Route 1-Direction: Downtown 8:59 a.m. bus trips. The Naive model ali. 54 ali. 2 boar 54 boar54 ali.54 m z k Table D9.31 -D9.45. Real-time Bus Arrival Prediction at Stops: Holmwood, Gladstone, and Rideau. 604.5 630.3 13 465.59 t6-R1-54 436.09 353.7 t6-R1-54 17 12 dcode dcode Table D9.31 Table D9.32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced Table D9.33 A r v . A r v . A r v . « ) V c t 1 t 2 i CO {2 . tc CO CO o t S i 2 tc CO QCO •Q <2 QCO *Q * * o © o <0 a \ dcode ali54 a Holmwood Gladatone Rideau Update t f i l m s i m sCM 0 o 3 >■ K CO CO CO o n i o - V CM « CO CO CD CM 1 t C CO n i T“ 2 t I M *© i o t c t i o t •Q *© i o c J T 1 CD CO X * T m ’ M i in 1 CO * + ■o *o 0 ( *2 Q»CO » ■Q © <0 - » o c © - o 5 O 3 © o O o o o © © • 1 1 • Update < d r.W jjjW i H . w :0 1 o i H 1 : 'p i m iiCM . m i i 2 H CO t I CO n i o t H CO CO T” o c «Q T M i s t I n i 4 *2 o c CC *o in -O M* o t ■o n i © 9 © © o C ? O 3 o © © O © o o o © © a a * 3 1 Update vi<« 0 1 i i 0 0 ■•csr ' t i H !**■ •*M gr ig - > s 0 > o 0 0 ▼» > o s CM 0 K - M CO CD CO oi CM MCM CM 0 0 o i O 00 - N § t I o c i o c IS 0 o c 1 1 CD :C :i n :jp e.CM . •* * ' CM 0 d © 0 N r M 1 0 0 s K S o c CO > o n i CM " T 0 ; » :;« 'h CO 0 o 0 0 CD I I last 1 >*• 0 d 0 - h GO ) O o o t S 0 o i o CO K >*• 0 00 K Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced ■'fr o o Table D9.36 < 5 . C > © > . . . . •O H CD W r o J T T> »p ' O CO «Q I 4 H o c CO DC IO 4 CO CO ■Q Arv. Arv. Arv. or CO o t CO 1 0 £ 5 1 CO 1 1 *© i CD ) f I « T“ Q • o c CO s CM Q • ? 5 1 1 if) - vJ 0 CO 0 re a a T 1 dcode a a I H o l m w o o d G l a d s t o n e R i d e a u U p d a t e :iiCp. * t f . m O I :; s m MO t f I i- //M *M '* o i /TTf ; Table D9.39 Arv. Arv. Arv. t C D CO 15 in CO s - I 3 "O co H CO tc m w "S M- in 1 <2 CO g (O o I o * ■ t dcode * Holmwood Gladstone Rideau Update . iM ■ym- "« ISO "T-' l«M 1 m S £ N CO CO CO w d CO CO Ot CO s CO o 04 00 O CM IO *» iC IO o> W to CD CM CM r* T“ CM CO r*- CM 3 £ m co 5 <0 3 4 12 ft: i *o» lip '>«* to CD o* CO CM o> CM S CO fw Y» CM o» CO O) CM s 3 ? ) V CO CM 5 CM CO ICO m lit <5 co T- CO CO in '■ (0 m CO Tf o 2 JK Q c« I I <1 s I 2o . E. . *0 •• {S V Of CO in « ■o ’O’ 15 CO 5 s ++ «Q 1 X 1- CO cc in CO I co ■o T> ■Q 3 1 +-• o * o c O CO o 5 o 3 O O o co * 0 « • ■ ■ Update PI It* IO I*? m CO «n in © o> CO CM o> o CO Table D9.42 Arv. Arv. Arv. ' O or T* tf> «0 co CD -Q i IO O CB *8 ■ ■ ■ dcode ■ dwel!54 I T6-R54-3(1) boar3 ali3 I dweH3 Holmwood Gladstone Rideau Update tt I ■M 88689 in in CM I 527.5 714.73 I 14.5 | 527.50 I 1256.73 | 1830.61 co I 499.95 I 14.5 I 5 i 8 ft 5 7 in co • n z 661.05 14.5 702.05 1175.50 I 1892.05 ** ft . N ft I I 642 i 8 844.4 2017.85 last I 649.5 499.95 1156.45 1822.95 H M OS 3 V es j j Arv. Arv. Arv. 1 CO O' r 4 -Q t- 7 CO in CO m ? CO CD ' O T* m CO ** - O <8 O C8 i ■ i • dcode 1 al!54 dwell 54__ ali3 dwel!3 Holmwood Gladstone Rideau Update ::^S4 i •cf N cm in CD 1 637.23 557.52 569.62 1 I 637.23 | 1206.75 | 1793.37 1 1st 1 DC CD CO CD iO CM CM 1 572 1 CO 5 t f t f CO s m ft a: co I I 578.25 14.5 | 526.27 1 I 1164.75 | 1705.52 2st \ 2 t f ft co CM CM 660 | m •5 <8 487.41 j 1755.91 579.5 572 1246.5 1848.00 -'T H a 3 ON 4> e* r—------Arv. Arv. Arv. £ •O t f CO m CO T“ tc s £2 I ■e i « ■e 5 i- CD in -Q CD ** 1 O <8 O <8 <8 1 ■ ■ * dcode 1 al!3 Holmwood Gladstone Rideau Update CO CO CD m 776.07 1 | . i § 8 i - 5 i 494.29 14 1 *♦.»! 4 I 603.25 776.07 1284.86 | 1902.61 1st I ■O’ 493 1 9.5 ___ 1 r- 3 ft t f co s m ft o: m co 1 i;:«» I*D D *0 00 ! ! 493.88 I CO 14.5 | 664.76 I 1675.64 f & tS IE jS lI ft 3 393 I h- K m CO ■ ■ _ 1 650.82 1563.32 1 /ast I 572.5 493 895.5 1 1485.60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. without prohibited reproduction Further owner. copyright the of permission with Reproduced o CO Table D9.45 Arv. Arv. Arv. 4 T~ •o H CO K IO CO £ co IO « X CO s V> «Q "5 JO >*■ 1 § •Q CO i £? o £ : P to IO IO Y» u> 1589.13 I CM co p IO CM o> O IO s 5 e p O to CO v> o» o o t CM IO p CO W> *b 1656.62 | co o> co T- p ) U £ § H> IO 1 •p y«0 o t 5 U> 1656.45 | o> ) O K p o t IO 01 0 (0 IO
Gladstone 1 Rideau Update ;.m vW ;.*■ !|0 “«* :;CO .ts* .yfm N W> R R CO ** to w CO m tf)